427 Sentences With "algebraically" | Random Sentence Generator

A third finds a futurist oracle algebraically predicting the day of someone's death.

Her beloved abstract surfaces can be described geometrically, with angles, lengths and areas, or algebraically, with equations.

And then you can also think of permutations algebraically—there's a natural sort of "multiplication" on permutations, in which the product of two permutations is the permutation you get by doing one permutation after the other.

Other shows include "Simple Math: Solving for the Neurobiology of Assault," an algebraically inclined play about an aspiring actress who discovers that her mentor is a sexual predator, written by Lisa Danielle Buch, and "The Existence Formula" from Sarah Rickman and the Juice Factory theater company, a comedic rumination on life and death.

Even assuming that every polynomial of the form xn − a splits into linear factors is not enough to assure that the field is algebraically closed. If a proposition which can be expressed in the language of first-order logic is true for an algebraically closed field, then it is true for every algebraically closed field with the same characteristic. Furthermore, if such a proposition is valid for an algebraically closed field with characteristic 0, then not only is it valid for all other algebraically closed fields with characteristic 0, but there is some natural number N such that the proposition is valid for every algebraically closed field with characteristic p when p > N.See subsections Rings and fields and Properties of mathematical theories in §2 of J. Barwise's "An introduction to first-order logic". Every field F has some extension which is algebraically closed.

As an example, the field of real numbers is not algebraically closed, because the polynomial equation x2 + 1 = 0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. Also, no finite field F is algebraically closed, because if a1, a2, ..., an are the elements of F, then the polynomial (x − a1)(x − a2) ··· (x − an) + 1 has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraically closed field is the field of (complex) algebraic numbers.

In a projective variety, a nonzero effective cycle has non- zero degree so cannot be algebraically equivalent to 0. In Hironaka's example the effective cycle consisting of the two exceptional curves is algebraically equivalent to 0.

Elasticity coefficients can be calculated in various ways, either numerically or algebraically.

A field is algebraically closed if it does not have any strictly bigger algebraic extensions or, equivalently, if any polynomial equation :, with coefficients , has a solution . By the fundamental theorem of algebra, is algebraically closed, i.e., any polynomial equation with complex coefficients has a complex solution. The rational and the real numbers are not algebraically closed since the equation : does not have any rational or real solution.

Basic invariants of a field include the characteristic and the transcendence degree of over its prime field. The latter is defined as the maximal number of elements in that are algebraically independent over the prime field. Two algebraically closed fields and are isomorphic precisely if these two data agree. This implies that any two uncountable algebraically closed fields of the same cardinality and the same characteristic are isomorphic.

Tsen's theorem is about the function field of an algebraic curve over an algebraically closed field.

With the exception of a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves to vector spaces over fields of characteristic zero. Because the theory of algebraically closed fields of characteristic zero is complete, a theory valid for a special algebraically closed field of characteristic zero is also valid for every other algebraically closed field of characteristic zero. Thus, without loss of generality, we can study vector spaces over \Complex.

In mathematics, algebraically compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution of infinite systems of equations in the module by finitary means. The solutions to these systems allow the extension of certain kinds of module homomorphisms. These algebraically compact modules are analogous to injective modules, where one can extend all module homomorphisms. All injective modules are algebraically compact, and the analogy between the two is made quite precise by a category embedding.

It is a consequence of Chevalley's theorem that finite fields are quasi-algebraically closed. This had been conjectured by Emil Artin in 1935. The motivation behind Artin's conjecture was his observation that quasi-algebraically closed fields have trivial Brauer group, together with the fact that finite fields have trivial Brauer group by Wedderburn's theorem.

Given a field extension , a subset S of L is called algebraically independent over K if no non-trivial polynomial relation with coefficients in K exists among the elements of S. The largest cardinality of an algebraically independent set is called the transcendence degree of L/K. It is always possible to find a set S, algebraically independent over K, such that L/K(S) is algebraic. Such a set S is called a transcendence basis of L/K. All transcendence bases have the same cardinality, equal to the transcendence degree of the extension.

The complexity of the problem is still open if it is assumed that all edge labels are unique and algebraically independent.

In Grothendieck's language, an abstract algebraic variety is usually defined to be an integral, separated scheme of finite type over an algebraically closed field, although some authors drop the irreducibility or the reducedness or the separateness condition or allow the underlying field to be not algebraically closed.Liu, Qing. Algebraic Geometry and Arithmetic Curves, p. 55 Definition 2.3.

47, and p. 88 Example 3.2.3 Classical algebraic varieties are the quasiprojective integral separated finite type schemes over an algebraically closed field.

The Diophantine dimension of a field is the smallest natural number k, if it exists, such that the field of is class Ck: that is, such that any homogeneous polynomial of degree d in N variables has a non-trivial zero whenever N > dk. Algebraically closed fields are of Diophantine dimension 0; quasi- algebraically closed fields of dimension 1. Clearly if a field is Ti then it is Ci, and T0 and C0 are equivalent, each being equivalent to being algebraically closed. It is not known whether Tsen rank and Diophantine dimension are equal in general.

Such an extension is called an algebraically closed extension. Among all such extensions there is one and only one (up to isomorphism, but not unique isomorphism) which is an algebraic extension of F;See Lang's Algebra, §VII.2 or van der Waerden's Algebra I, §10.1. it is called the algebraic closure of F. The theory of algebraically closed fields has quantifier elimination.

A connected linear algebraic group G over an algebraically closed field is called semisimple if every smooth connected solvable normal subgroup of G is trivial. More generally, a connected linear algebraic group G over an algebraically closed field is called reductive if the largest smooth connected unipotent normal subgroup of G is trivial.SGA 3 (2011), v. 3, Définition XIX.1.6.1.

Throughout the example section, the base field is algebraically closed and has characteristic zero. All the examples below (except the first one) are from .

In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .

A Poincaré space is a topological space whose singular chain complex is a Poincaré complex. These are used in surgery theory to analyze manifold algebraically.

If a field is Ci then so is a finite extension.Lang (1997) p.245 The C0 fields are precisely the algebraically closed fields.Lorenz (2008) p.

Given a field extension L/K, Zorn's lemma can be used to show that there always exists a maximal algebraically independent subset of L over K. Further, all the maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension. For every finite set S of elements of L, the algebraically independent subsets of S satisfy the axioms that define the independent sets of a matroid. In this matroid, the rank of a set of elements is its transcendence degree, and the flat generated by a set T of elements is the intersection of L with the field K[T].Oxley (1992) p.

Over an algebraically closed field, any matrix A has a Jordan normal form and therefore admits a basis of generalized eigenvectors and a decomposition into generalized eigenspaces.

A connected linear algebraic group G over an algebraically closed field is called semisimple if every smooth connected solvable normal subgroup of G is trivial. More generally, a connected linear algebraic group G over an algebraically closed field is called reductive if every smooth connected unipotent normal subgroup of G is trivial.Milne (2017), Definition 6.46. (Some authors do not require reductive groups to be connected.) A semisimple group is reductive.

Independently of the range of the interaction, at low enough temperature the magnetization is positive. Conjecturally, in each of the low temperature extremal states the truncated correlations decay algebraically.

These problems are easy for k algebraically closed, and they are understood for some other fields such as number fields, but for arbitrary fields there are many open questions.

PageRank can be computed either iteratively or algebraically. The iterative method can be viewed as the power iteration method or the power method. The basic mathematical operations performed are identical.

The field F is algebraically closed if and only if every polynomial p(x) of degree n ≥ 1, with coefficients in F, splits into linear factors. In other words, there are elements k, x1, x2, ..., xn of the field F such that p(x) = k(x − x1)(x − x2) ··· (x − xn). If F has this property, then clearly every non-constant polynomial in F[x] has some root in F; in other words, F is algebraically closed. On the other hand, that the property stated here holds for F if F is algebraically closed follows from the previous property together with the fact that, for any field K, any polynomial in K[x] can be written as a product of irreducible polynomials.

In algebra, the real radical of an ideal I in a polynomial ring with real coefficients is the largest ideal containing I with the same vanishing locus. It plays a similar role in real algebraic geometry that the radical of an ideal plays in algebraic geometry over an algebraically closed field. More specifically, the Nullstellensatz says that when I is an ideal in a polynomial ring with coefficients coming from an algebraically closed field, the radical of I is the set of polynomials vanishing on the vanishing locus of I. In real algebraic geometry, the Nullstellensatz fails as the real numbers are not algebraically closed. However, one can recover a similar theorem, the real Nullstellensatz, by using the real radical in place of the (ordinary) radical.

The field F is algebraically closed if and only if, for each natural number n, every linear map from Fn into itself has some eigenvector. An endomorphism of Fn has an eigenvector if and only if its characteristic polynomial has some root. Therefore, when F is algebraically closed, every endomorphism of Fn has some eigenvector. On the other hand, if every endomorphism of Fn has an eigenvector, let p(x) be an element of F[x].

Unlike palindromes, it is also font dependent. The concept of strobogrammatic numbers is not neatly expressible algebraically, the way that the concept of repunits is, or even the concept of palindromic numbers.

The notation allows moving boundary conditions of summations (or integrals) as a separate factor into the summand, freeing up space around the summation operator, but more importantly allowing it to be manipulated algebraically.

Laurent series cannot in general be multiplied. Algebraically, the expression for the terms of the product may involve infinite sums which need not converge (one cannot take the convolution of integer sequences). Geometrically, the two Laurent series may have non-overlapping annuli of convergence. Two Laurent series with only finitely many negative terms can be multiplied: algebraically, the sums are all finite; geometrically, these have poles at c, and inner radius of convergence 0, so they both converge on an overlapping annulus.

The Goldberg–Sachs theorem is a result in Einstein's theory of general relativity about vacuum solutions of the Einstein field equations relating the existence of a certain type of congruence with algebraic properties of the Weyl tensor. More precisely, the theorem states that a vacuum solution of the Einstein field equations will admit a shear-free null geodesic congruence if and only if the Weyl tensor is algebraically special. The theorem is often used when searching for algebraically special vacuum solutions.

Many results for diagonalizable matrices hold only over an algebraically closed field (such as the complex numbers). In this case, diagonalizable matrices are dense in the space of all matrices, which means any defective matrix can be deformed into a diagonalizable matrix by a small perturbation; and the Jordan normal form theorem states that any matrix is uniquely the sum of a diagonalizable matrix and a nilpotent matrix. Over an algebraically closed field, diagonalizable matrices are equivalent to semi-simple matrices.

A smooth connected unipotent group over a perfect field k (for example, an algebraically closed field) has a composition series with all quotient groups isomorphic to the additive group Ga.Borel (1991), Theorem 15.4(iii).

To put it algebraically, 2n + 1 = p + 2q always has a solution in primes p and q (not necessarily distinct) for n > 2. The Lemoine conjecture is similar to but stronger than Goldbach's weak conjecture.

The classical objects of interest in arithmetic geometry are rational points: sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or function fields, i.e. fields that are not algebraically closed excluding the real numbers. Rational points can be directly characterized by height functions which measure their arithmetic complexity. The structure of algebraic varieties defined over non-algebraically-closed fields has become a central area of interest that arose with the modern abstract development of algebraic geometry.

If F is an algebraically closed field and n is a natural number, then F contains all nth roots of unity, because these are (by definition) the n (not necessarily distinct) zeroes of the polynomial xn − 1\. A field extension that is contained in an extension generated by the roots of unity is a cyclotomic extension, and the extension of a field generated by all roots of unity is sometimes called its cyclotomic closure. Thus algebraically closed fields are cyclotomically closed. The converse is not true.

Algebraically eliminate the immaterial constant B from these two equations to deduce that all initial conditions y_0+y'_0=1+A have the same A, hence the same long term evolution, and hence form an isochron.

203] Over an algebraically closed field K (for example the complex numbers C), there are no finite-dimensional associative division algebras, except K itself.Cohn (2003), [ Proposition 5.4.5, p. 150] Associative division algebras have no zero divisors.

Here is the product of the distinct irreducible factors of . For algebraically closed fields it is the polynomial of minimum degree that has the same roots as ; in this case gives the number of distinct roots of .

It is estimated that 500 million shapes can be defined algebraically in four dimensions, and a few thousand more in the fifth. The project has already won the Philip Leverhulme Prize—worth £70,000—from the Leverhulme Trust.

Over an algebraically closed field of characteristic 0, any unipotent abelian connected algebraic group is isomorphic to a product of copies of the additive group G_a. The analogue of this for fields of characteristic p is false: the truncated Witt schemes are counterexamples. (We make them into algebraic groups by forgetting the multiplication and just using the additive structure.) However, these are essentially the only counterexamples: over an algebraically closed field of characteristic p, any unipotent abelian connected algebraic group is isogenous to a product of truncated Witt group schemes.

A surcomplex number is a number of the form , where a and b are surreal numbers and is the square root of .Surreal vectors and the game of Cutblock, James Propp, August 22, 1994. The surcomplex numbers form an algebraically closed field (except for being a proper class), isomorphic to the algebraic closure of the field generated by extending the rational numbers by a proper class of algebraically independent transcendental elements. Up to field isomorphism, this fact characterizes the field of surcomplex numbers within any fixed set theory.

Many other transcendental numbers remain unclassified. Two numbers x, y are called algebraically dependent if there is a non-zero polynomial P in 2 indeterminates with integer coefficients such that P(x, y) = 0\. There is a powerful theorem that 2 complex numbers that are algebraically dependent belong to the same Mahler class.. This allows construction of new transcendental numbers, such as the sum of a Liouville number with e or . The symbol S probably stood for the name of Mahler's teacher Carl Ludwig Siegel, and T and U are just the next two letters.

The basic definitions and facts above enable one to do classical algebraic geometry. To be able to do more -- for example, to deal with varieties over fields that are not algebraically closed -- some foundational changes are required. The modern notion of a variety is considerably more abstract than the one above, though equivalent in the case of varieties over algebraically closed fields. An abstract algebraic variety is a particular kind of scheme; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings.

E-functions were first studied by Siegel in 1929.C.L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. 1, 1929. He found a method to show that the values taken by certain E-functions were algebraically independent.

A lattice is a partially ordered set that is both a meet- and join-semilattice with respect to the same partial order. Algebraically, a lattice is a set with two associative, commutative idempotent binary operations linked by corresponding absorption laws.

It is the union of the finite fields containing (the ones of order ). For any algebraically closed field of characteristic 0, the algebraic closure of the field of Laurent series is the field of Puiseux series, obtained by adjoining roots of .

The L-groups are defined algebraically in terms of quadratic forms or in terms of chain complexes with quadratic structure. See the main article for more details. Here only the properties of the L-groups described below will be important.

The uniqueness of the representation can be proved inductively in a similar way. (It is equivalent to the fact that the polynomials are algebraically independent over the ring .) The fact that the polynomial representation is unique implies that is isomorphic to .

The Riemann tensor has only one algebraically independent, nonzero component :R_{0202}=-1, which shows that the spacetime is Ricci flat but not conformally flat. That is sufficient to conclude that it is a vacuum solution distinct from Minkowski spacetime.

Unipotent groups over an algebraically closed field of any given dimension can in principle be classified, but in practice the complexity of the classification increases very rapidly with the dimension, so people tend to give up somewhere around dimension 6.

Let B be a ring that is integral over a subring A and k an algebraically closed field. If f: A \to k is a homomorphism, then f extends to a homomorphism B → k. This follows from the going-up.

Any set of polynomials may be viewed as a system of polynomial equations by equating the polynomials to zero. The set of the solutions of such a system depends only on the generated ideal, and, therefore does not change when the given generating set is replaced by the Gröbner basis, for any ordering, of the generated ideal. Such a solution, with coordinates in an algebraically closed field containing the coefficients of the polynomials, is called a zero of the ideal. In the usual case of rational coefficients, this algebraically closed field is chosen as the complex field.

A careful distinction needs to be made between abstract quantities and measurable quantities. The multiplication and division rules of quantity calculus are applied to SI base units (which are measurable quantities) to define SI derived units, including dimensionless derived units, such as the radian (rad) and steradian (sr) which are useful for clarity, although they are both algebraically equal to 1. Thus there is some disagreement about whether it is meaningful to multiply or divide units. Emerson suggests that if the units of a quantity are algebraically simplified, they then are no longer units of that quantity.

A smooth connected curve over an algebraically closed field is called hyperbolic if 2g-2+r>0 where g is the genus of the smooth completion and r is the number of added points. Over an algebraically closed field of characteristic 0, the fundamental group of X is free with 2g+r-1 generators if r>0. (Analogue of Dirichlet's unit theorem) Let X be a smooth connected curve over a finite field. Then the units of the ring of regular functions O(X) on X is a finitely generated abelian group of rank r -1.

In the context of algebraic geometry, the statement applies for smooth and proper varieties over an algebraically closed field. This variant of the Atiyah–Bott fixed point formula was proved by by expressing both sides of the formula as appropriately chosen categorical traces.

It has been shown by Dain and Moreschi that a corresponding theorem will not hold in linearized gravity, that is, given a solution of the linearised Einstein field equations admitting a shear-free null congruence, then this solution need not be algebraically special.

If F is not algebraically closed, let p(x) be a polynomial whose degree is at least 1 without roots. Then p(x) and p(x) are not relatively prime, but they have no common roots (since none of them has roots).

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21M. F. Atiyah and I. G. Macdonald (1969).

Bulletin of Mathematical Biology, 56(1):1–64, 1994. The model was presented at the second International Conference of Artificial Life. In his first papers he presented the concept of organization, as a set of molecules that is algebraically closed and self-maintaining.

There is a systematic way to choose generators for a peripheral subgroup of a knot in 3-space, such that distinct knot types always have algebraically distinct peripheral systems. The generators in this situation are called a longitude and a meridian of the knot complement.

Gauss deduced that a regular -gon can be constructed if . Building on Lagrange's work, Paolo Ruffini claimed (1799) that quintic equations (polynomial equations of degree 5) cannot be solved algebraically; however, his arguments were flawed. These gaps were filled by Niels Henrik Abel in 1824.

Many distributions are defined with mathematically equivalent but algebraically different formulas. This leads to issues when exchanging models between software tools. LeBauer DS et al. Translating probability density functions: From R to BUGS and back again, R Journal, 2013 The following examples illustrate that.

These sets can be given a natural topological structure: the Zariski–Riemann space of K/k. In case k is algebraically closed, the Zariski-Riemann space of K/k is a smooth curve over k and K is the function field of this curve.

Ten15 is an algebraically specified abstract machine. It was developed by Foster, Currie et al. at the Royal Signals and Radar Establishment at Malvern, Worcestershire, during the 1980s. It arose from earlier work on the Flex machine, which was a capability computer implemented via microcode.

In mathematics, the Noether inequality, named after Max Noether, is a property of compact minimal complex surfaces that restricts the topological type of the underlying topological 4-manifold. It holds more generally for minimal projective surfaces of general type over an algebraically closed field.

Example: In the theory of algebraically closed fields of characteristic 0, there is a 1-type represented by elements that are transcendental over the prime field. This is a non-isolated point of the Stone space (in fact, the only non-isolated point). The field of algebraic numbers is a model omitting this type, and the algebraic closure of any transcendental extension of the rationals is a model realizing this type. All the other types are "algebraic numbers" (more precisely, they are the sets of first-order statements satisfied by some given algebraic number), and all such types are realized in all algebraically closed fields of characteristic 0.

In the twentieth century, the Lefschetz principle, named for Solomon Lefschetz, was cited in algebraic geometry to justify the use of topological techniques for algebraic geometry over any algebraically closed field K of characteristic 0, by treating K as if it were the complex number field. An elementary form of it asserts that true statements of the first order theory of fields about C are true for any algebraically closed field K of characteristic zero. A precise principle and its proof are due to Alfred Tarski and are based in mathematical logic.For discussions see Abraham Seidenberg, Comments on Lefschetz's Principle, American Mathematical Monthly, Vol.

There is also a similar notion: a field extension L / k is said to be self-regular if L \otimes_k L is an integral domain. A self-regular extension is relatively algebraically closed in k.Cohn (2003) p.427 However, a self-regular extension is not necessarily regular.

All the above conjectures and theorems are consequences of the unproven extension of Baker's theorem, that logarithms of algebraic numbers that are linearly independent over the rational numbers are automatically algebraically independent too. The diagram on the right shows the logical implications between all these results.

Since the results are given for nth order, a computer program can be developed which will give the results for mode shapes and natural frequencies to the desired accuracy, preempting the need to go through the mathematically arduous task of deriving the higher order expressions algebraically.

Current methods in computational nonlinear algebra can be broadly broken into two domains: symbolic and numerical. Symbolic methods often rely on the computation of Gröbner bases. On the other hand, numerical methods typically use algebraically-founded homotopy continuation, with a base field of the complex numbers.

216 A matroid that can be generated in this way is called algebraic or algebraically representable.Oxley (1992) p.218 No good characterization of algebraic matroids is known,Oxley (1992) p.215 but certain matroids are known to be non-algebraic; the smallest is the Vámos matroid..

Then Hrushovski and Zilber prove that under these conditions there is an algebraically closed field K, and a non-singular algebraic curve C, such that its Zariski geometry of powers and their Zariski topology is isomorphic to the given one. In short, the geometry can be algebraized.

Formally, let X be a variety defined over an algebraically closed field of characteristic zero: hence X is defined over a finitely generated field E. If the set of points X(F) is finite for any finitely generated field extension F of E, then X is Mordellic.

For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field.

In mathematics, in the realm of group theory, a group A\ is algebraically closed if any finite set of equations and inequations that "make sense" in A\ have a solution in A\ without needing a group extension. This notion will be made precise later in the article in .

For example, is : (x_1 + 3x_2 - x_3)(3x_1 + x_4 - 1) \cdots (x_7 - x_2) \equiv 0\ ? To solve this, we can multiply it out and check that all the coefficients are 0\. However, this takes exponential time. In general, a polynomial can be algebraically represented by an arithmetic formula or circuit.

Algebraically, hyperbolic and spherical geometry have the same structure. This allows us to apply concepts and theorems to one geometry to the other. Applying hyperbolic geometry to spherical geometry can make it easier to understand because spheres are much more concrete, which then makes spherical geometry easier to conceptualize.

In general the approach of taking the torsor theory, easy over an algebraically closed field, and trying to get back 'down' to a smaller field is an aspect of descent. It leads at once to questions of Galois cohomology, since the torsors represent classes in group cohomology H1.

For bonds with multiple coupons, it is not generally possible to solve for yield in terms of price algebraically. A numerical root-finding technique such as Newton's method must be used to approximate the yield, which renders the present value of future cash flows equal to the bond price.

However, accurate quantum- mechanical calculations (starting in the 1970s) have shown that the reason is that the electrons in singly occupied orbitals are less effectively screened or shielded from the nucleus, so that such orbitals contract and electron–nucleus attraction energy becomes greater in magnitude (or decreases algebraically).

In other words, the only linear transformations of M that commute with all transformations coming from R are scalar multiples of the identity. This holds more generally for any algebra R over an uncountable algebraically closed field k and for any simple module M that is at most countably-dimensional: the only linear transformations of M that commute with all transformations coming from R are scalar multiples of the identity. When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is still of particular interest. A simple module over k-algebra is said to be absolutely simple if its endomorphism ring is isomorphic to k.

The choice of the origin is arbitrary: any other point may be chosen, as the representation is of an affine space. The origin merely represents a reference point, and is algebraically equivalent to any other point. As with any translation, changing the origin corresponds to a rotation in the representation space.

The field C has the following three properties: first, it has characteristic 0. This means that for any number of summands (all of which equal one). Second, its transcendence degree over Q, the prime field of C, is the cardinality of the continuum. Third, it is algebraically closed (see above).

Computing roots of functions, for example polynomial functions, frequently requires the use of specialised or approximation techniques (e.g., Newton's method). However, some polynomial functions, including all those of degree no greater than 4, can have all their roots expressed algebraically in terms of their coefficients (for more, see algebraic solution).

The projectivities are described algebraically as homographies, since the real numbers form a ring, according to the general construction of a projective line over a ring. Collectively they form the group PGL(2,R). The projectivities which are their own inverses are called involutions. A hyperbolic involution has two fixed points.

This fact has had unexpected applications to neural network learning.See Anthony and Bartlett for details. Examples of NIP theories include also the theories of all the following structures:See Simon, Appendix A. linear orders, trees, abelian linearly ordered groups, algebraically closed valued fields, and the p-adic field for any p.

Linear systems of equations may be solved algebraically by isolating one of the variables and then performing a substitution. Isolating a variable can be modeled with algebra tiles in a manner similar to solving linear equations (above), and substitution can be modeled with algebra tiles by replacing tiles with other tiles.

Quasi-algebraically closed fields are also called C1. A Ck field, more generally, is one for which any homogeneous polynomial of degree d in N variables has a non-trivial zero, provided :dk < N, for k ≥ 1.Serre (1997) p.87 The condition was first introduced and studied by Lang.

In other words, a reasonable increase in elements (braid twists) can achieve a high degree of accuracy in the answer. Actual computation [gates] are done by the edge states of a fractional quantum Hall effect. This makes models of one-dimensional anyons important. In one space dimension, anyons are defined algebraically.

Boris Zilber conjectured that the only pregeometries that can arise from strongly minimal sets are those that arise in vector spaces, projective spaces, or algebraically closed fields. This conjecture was refuted by Ehud Hrushovski, who developed a method known as the "Hrushovski construction" to build new strongly minimal structures from finite structures.

As a group, the Jacobian variety of a curve is isomorphic to the quotient of the group of divisors of degree zero by the subgroup of principal divisors, i.e., divisors of rational functions. This holds for fields that are not algebraically closed, provided one considers divisors and functions defined over that field.

A point at infinity of the curve is a zero of p of the form (a, b, 0). Equivalently, (a, b) is a zero of pd. The fundamental theorem of algebra implies that, over an algebraically closed field (typically, the field of complex numbers), pd factors into a product of linear factors.

An elementary number is one formalization of the concept of a closed-form number. The elementary numbers form an algebraically closed field containing the roots of arbitrary equations using field operations, exponentiation, and logarithms. The set of the elementary numbers is subdivided into the explicit elementary numbers and the implicit elementary numbers.

Unlike the complex numbers, the split-complex numbers are not algebraically closed, and further contain nontrivial zero divisors and non-trivial idempotents. As with the quaternions, split-quaternions are not commutative, but further contain nilpotents; they are isomorphic to the 2 × 2 real matrices. Split-octonions are non-associative and contain nilpotents.

It appears that it also plays a role that near a change of the tendency (e.g. from falling to rising prices) there are typical "panic reactions" of the selling or buying agents with algebraically increasing bargain rapidities and volumes.See for example Preis, Mantegna, 2003. The "fat tails" are also observed in commodity markets.

Often, it is difficult or impossible to solve explicitly for y, and implicit differentiation is the only feasible method of differentiation. An example is the equation :y^5-y=x. It is impossible to algebraically express y explicitly as a function of x, and therefore one cannot find dy/dx by explicit differentiation.

Image for 9-points theorem, special case, when both and are unions of 3 lines In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane . The original form states: :Assume that two cubics and in the projective plane meet in nine (different) points, as they do in general over an algebraically closed field. Then every cubic that passes through any eight of the points also passes through the ninth point. A more intrinsic form of the Cayley–Bacharach theorem reads as follows: :Every cubic curve on an algebraically closed field that passes through a given set of eight points also passes through a certain (fixed) ninth point , counting multiplicities.

Bedford and Cooke show that any assignment of values in the open interval (−1, 1) to the edges in any partial correlation vine is consistent, the assignments are algebraically independent, and there is a one- to-one relation between all such assignments and the set of correlation matrices. In other words, partial correlation vines provide an algebraically independent parametrization of the set of correlation matrices, whose terms have an intuitive interpretation. Moreover, the determinant of the correlation matrix is the product over the edges of (1 − ρ2ik;D(ik)) where ρik;D(ik) is the partial correlation assigned to the edge with conditioned variables i,k and conditioning variables D(ik). A similar decomposition characterizes the mutual information, which generalizes the determinant of the correlation matrix.

The special set for a projective variety V is the Zariski closure of the union of the images of all non-trivial maps from algebraic groups into V. Lang conjectured that the complement of the special set is Mordellic. A variety is algebraically hyperbolic if the special set is empty. Lang conjectured that a variety X is Mordellic if and only if X is algebraically hyperbolic and that this is turn equivalent to X being pseudo-canonical. For a complex algebraic variety X we similarly define the analytic special or exceptional set as the Zariski closure of the union of images of non-trivial holomorphic maps from C to X. Brody's definition of a hyperbolic variety is that there are no such maps.

A vacuum metric, R_{ab}=0, is algebraically special if and only if it contains a shear-free null geodesic congruence; the tangent vector obeys k_{[a}C_{b]ijc}k^ik^j=0.; originally published in Acta Phys. Pol. 22, 13–23 (1962). This is the theorem originally stated by Goldberg and Sachs.

In mathematics, a compactly generated (topological) group is a topological group G which is algebraically generated by one of its compact subsets.. This should not be confused with the unrelated notion (widely used in algebraic topology) of a compactly generated space -- one whose topology is generated (in a suitable sense) by its compact subspaces.

This is sometimes called the large Cartan subgroup. There is also a small Cartan subgroup, defined to be the centralizer of a maximal torus. These Cartan subgroups need not be abelian in general. For connected algebraic groups over an algebraically closed field a Cartan subgroup is usually defined as the centralizer of a maximal torus.

An alternative to the White test is the Breusch–Pagan test, where the Breusch-Pagan test is designed to detect only linear forms of heteroskedasticity. Under certain conditions and a modification of one of the tests, they can be found to be algebraically equivalent. If homoskedasticity is rejected one can use heteroskedasticity- consistent standard errors.

An example is the field of complex numbers. Every field has an algebraic extension which is algebraically closed (called its algebraic closure), but proving this in general requires some form of the axiom of choice. An extension L/K is algebraic if and only if every sub K-algebra of L is a field.

The difference lies in that a vector manifold is algebraically rich while a manifold is not. Since this is the primary motivation for vector manifolds the following interpretation is rewarding. Consider a vector manifold as a special set of "points". These points are members of an algebra and so can be added and multiplied.

Generically (if a certain linear function of electromagnetic field does not vanish identically), three out of four components of the spinor function in the Dirac equation can be algebraically eliminated, yielding an equivalent fourth-order partial differential equation for just one component. Furthermore, this remaining component can be made real by a gauge transform.

That the Weyr form is a canonical form of a matrix is a consequence of the following result: Each square matrix A over an algebraically closed field is similar to a Weyr matrix W which is unique up to permutation of its basic blocks. The matrix W is called the Weyr (canonical) form of A.

Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and squaring the circle cannot be done with a compass and straightedge. Moreover, it shows that quintic equations are algebraically unsolvable. Fields serve as foundational notions in several mathematical domains.

The singularities of a curve are not birational invariants. However, locating and classifying the singularities of a curve is one way of computing the genus, which is a birational invariant. For this to work, we should consider the curve projectively and require F to be algebraically closed, so that all the singularities which belong to the curve are considered.

The Levi decomposition expresses an arbitrary Lie algebra as a semidirect sum of its solvable radical and a semisimple Lie algebra, almost in a canonical way. (Such a decomposition exists for a finite-dimensional Lie algebra over a field of characteristic zero.) Furthermore, semisimple Lie algebras over an algebraically closed field have been completely classified through their root systems.

First, if F is not algebraically closed, then isotropic subspaces may not exist: for a general theory, one needs to use the split orthogonal groups. Second, for vector spaces of even dimension 2m, isotropic subspaces of dimension m come in two flavours ("self-dual" and "anti-self- dual") and one needs to distinguish these to obtain a homogeneous space.

A parody version recorded by Mickey Katz is entitled "The Baby, the Bubbe, and You". The lyrics of the song are algebraically analyzed in a math lesson by Mr. Garrison in the South Park episode "Royal Pudding". The title of the song is sung several times in Buck-Tick's song "Django!!! -Genwaku no Django-" from their album Razzle Dazzle.

A scheme X is said to be generically smooth of dimension n over k if X contains an open dense subset that is smooth of dimension n over k. Every variety over a perfect field (in particular an algebraically closed field) is generically smooth.Lemma 1 in section 28 and Corollary to Theorem 30.5, Matsumura, Commutative Ring Theory (1989).

Illustrative example: Suppose M is an algebraically closed field. The theory has quantifier elimination . This allows us to show that a type is determined exactly by the polynomial equations it contains. Thus the space of n-types over a subfield A is bijective with the set of prime ideals of the polynomial ring A[x_1,\ldots,x_n].

Another phenomenon which can only occur in positive characteristic is that a K3 surface may be unirational. Michael Artin observed that every unirational K3 surface over an algebraically closed field must have Picard number 22. (In particular, a unirational K3 surface must be supersingular.) Conversely, Artin conjectured that every K3 surface with Picard number 22 must be unirational.

If a, b, c, d... represent the decimal odds, i.e. (fractional odds + 1), then an 'odds multiplier' OM can be calculated algebraically by multiplying the expressions (a + 1), (b + 1), (c + 1)... etc. together in the required manner and adding or subtracting additional components. If required, (decimal odds + 1) may be replaced by (fractional odds + 2).

In mathematics, two functions have a contact of order k if, at a point P, they have the same value and k equal derivatives. This is an equivalence relation, whose equivalence classes are generally called jets. The point of osculation is also called the double cusp. Contact is a geometric notion; it can be defined algebraically as a valuation.

In number theory, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by . Chevalley's theorem implied Artin's and Dickson's conjecture that finite fields are quasi-algebraically closed fields .

In mathematical physics, Kundt spacetimes are Lorentzian manifolds admitting a geodesic null congruence with vanishing optical scalars (expansion, twist and shear). A well known member of Kundt class is pp-wave. Ricci-flat Kundt spacetimes in arbitrary dimension are algebraically special. In four dimensions Ricci-flat Kundt metrics of Petrov type III and N are completely known.

Bootstrap:Algebra is taught in the teaching subsets of the Racket programming language, and Bootstrap:Reactive, Bootstrap: Data Science, and Bootstrap:Physics move students to Pyret. Both are functional languages, meaning they behave algebraically and so are well-suited to a math class. Bootstrap students primarily use cloud-based programming environments-- WeScheme for Bootstrap:Algebra and code.pyret.org for Bootstrap:Reactive, Bootstraps:Data Science, and Bootstrap:Physics.

Furthermore, the four base points determine three line pairs (degenerate conics through the base points, each line of the pair containing exactly two base points) and so each pencil of conics will contain at most three degenerate conics.. A pencil of conics can be represented algebraically in the following way. Let and be two distinct conics in a projective plane defined over an algebraically closed field . For every pair of elements of , not both zero, the expression: ::\lambda C_1 + \mu C_2 represents a conic in the pencil determined by and . This symbolic representation can be made concrete with a slight abuse of notation (using the same notation to denote the object as well as the equation defining the object.) Thinking of , say, as a ternary quadratic form, then is the equation of the "conic ".

In mathematics, a field F is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial P over F has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether, in a 1936 paper ; and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper . The idea itself is attributed to Lang's advisor Emil Artin. Formally, if P is a non-constant homogeneous polynomial in variables :X1, ..., XN, and of degree d satisfying :d < N then it has a non-trivial zero over F; that is, for some xi in F, not all 0, we have :P(x1, ..., xN) = 0.

In general the determination of an intersection leads to non-linear equations, which can be solved numerically, for example using Newton iteration. Intersection problems between a line and a conic section (circle, ellipse, parabola, etc.) or a quadric (sphere, cylinder, hyperboloid, etc.) lead to quadratic equations that can be easily solved. Intersections between quadrics lead to quartic equations that can be solved algebraically.

Finally if G is a divisible group and R is a real closed field, then R((G)) is a real closed field, and if R is algebraically closed, then so is R((G)). This theory is due to Hans Hahn, who also showed that one obtains subfields when the number of (non-zero) terms is bounded by some fixed infinite cardinality.

According to Bézout's theorem, two different cubic curves over an algebraically closed field which have no common irreducible component meet in exactly nine points (counted with multiplicity). The Cayley-Bacharach theorem thus asserts that the last point of intersection of any two members in the family of curves does not move if eight intersection points (without seven co-conic ones) are already prescribed.

The triangles with reflection symmetry are isosceles, the quadrilaterals with this symmetry are kites and isosceles trapezoids. For each line or plane of reflection, the symmetry group is isomorphic with Cs (see point groups in three dimensions for more), one of the three types of order two (involutions), hence algebraically isomorphic to C2. The fundamental domain is a half-plane or half-space.

The Dieudonné–Manin classification theorem was proved by and . It describes the structure of Dieudonné modules over an algebraically closed field k up to "isogeny". More precisely, it classifies the finitely generated modules over D_k[1/p], where D_k is the Dieudonné ring. The category of such modules is semisimple, so every module is a direct sum of simple modules.

In diophantine geometry the characteristic problems of the subject are those caused by the fact that the ground field K is not taken to be algebraically closed. The field of definition of a variety given abstractly may be smaller than the ground field, and two varieties may become isomorphic when the ground field is enlarged, a major topic in Galois cohomology.

A matrix or, equivalently, a linear operator T on a finite- dimensional vector space V is called semi-simple if every T-invariant subspace has a complementary T-invariant subspace.Lam (2001), [ p. 39] This is equivalent to the minimal polynomial of T being square-free. For vector spaces over an algebraically closed field F, semi-simplicity of a matrix is equivalent to diagonalizability.

Algebraic curves are the curves considered in algebraic geometry. A plane algebraic curve is the set of the points of coordinates such that , where is a polynomial in two variables defined over some field . One says that the curve is defined over . Algebraic geometry normally considers not only points with coordinates in but all the points with coordinates in an algebraically closed field .

The Dieudonné–Manin classification theorem was proved by and . It describes the structure of F-isocrystals over an algebraically closed field k. The category of such F-isocrystals is abelian and semisimple, so every F-isocrystal is a direct sum of simple F-isocrystals. The simple F-isocrystals are the modules Es/r where r and s are coprime integers with r>0.

The Levi-Civita field is similar to the Laurent series, but is algebraically closed. For example, the basic infinitesimal x has a square root. This field is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented in floating point.

There are other proofs of the theorem. Armand Borel gave a proof using topology. The case of n = 1 and field C follows since C is algebraically closed and can also be thought of as a special case of the result that for any analytic function f on C, injectivity of f implies surjectivity of f. This is a corollary of Picard's theorem.

65, No. 9 (Nov., 1958), pp. 685–690; 'Gerhard Frey and Hans-Georg Rück, The strong Lefschetz principle in algebraic geometry, Manuscripta Mathematica, Volume 55, Numbers 3–4, September, 1986, pp. 385–401. This principle permits the carrying over of some results obtained using analytic or topological methods for algebraic varieties over C to other algebraically closed ground fields of characteristic 0.

In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first-order language of fields that is true for the complex numbers is also true for any algebraically closed field of characteristic 0.

Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces.

The use of channels for communication is one of the features distinguishing the process calculi from other models of concurrency, such as Petri nets and the actor model (see Actor model and process calculi). One of the fundamental motivations for including channels in the process calculi was to enable certain algebraic techniques, thereby making it easier to reason about processes algebraically.

A Diophantine equation is a (usually multivariate) polynomial equation with integer coefficients for which one is interested in the integer solutions. Algebraic geometry is the study of the solutions in an algebraically closed field of multivariate polynomial equations. Two equations are equivalent if they have the same set of solutions. In particular the equation P = Q is equivalent to P-Q = 0.

In differential geometry and general relativity, the Bach tensor is a trace- free tensor of rank 2 which is conformally invariant in dimension .Rudolf Bach, "Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs", Mathematische Zeitschrift, 9 (1921) pp. 110. Before 1968, it was the only known conformally invariant tensor that is algebraically independent of the Weyl tensor.P. Szekeres, Conformal Tensors.

Let k be an algebraically closed field and G a linear algebraic group (that is, affine algebraic group) over k. By definition, Lie(G) is the Lie algebra of all derivations of k[G] that commute with the left action of G. As in the Lie group case, it can be identified with the tangent space to G at the identity element.

For any field F, if two polynomials p(x),q(x) ∈ F[x] are relatively prime then they do not have a common root, for if a ∈ F was a common root, then p(x) and q(x) would both be multiples of x − a and therefore they would not be relatively prime. The fields for which the reverse implication holds (that is, the fields such that whenever two polynomials have no common root then they are relatively prime) are precisely the algebraically closed fields. If the field F is algebraically closed, let p(x) and q(x) be two polynomials which are not relatively prime and let r(x) be their greatest common divisor. Then, since r(x) is not constant, it will have some root a, which will be then a common root of p(x) and q(x).

For a scheme X that is of finite type over C, the complex numbers, there is a close relation between the étale fundamental group of X and the usual, topological, fundamental group of X(C), the complex analytic space attached to X. The algebraic fundamental group, as it is typically called in this case, is the profinite completion of π1(X). This is a consequence of the Riemann existence theorem, which says that all finite étale coverings of X(C) stem from ones of X. In particular, as the fundamental group of smooth curves over C (i.e., open Riemann surfaces) is well understood; this determines the algebraic fundamental group. More generally, the fundamental group of a proper scheme over any algebraically closed field of characteristic zero is known, because an extension of algebraically closed fields induces isomorphic fundamental groups.

Once again, the field extension discussed above is a key example: if is not algebraic (i.e., is not a root of a polynomial with coefficients in ), then is isomorphic to . This isomorphism is obtained by substituting to in rational fractions. A subset of a field is a transcendence basis if it is algebraically independent (don't satisfy any polynomial relations) over and if is an algebraic extension of .

In model theory, a branch of mathematical logic, a C-minimal theory is a theory that is "minimal" with respect to a ternary relation C with certain properties. Algebraically closed fields with a (Krull) valuation are perhaps the most important example. This notion was defined in analogy to the o-minimal theories, which are "minimal" (in the same sense) with respect to a linear order.

Mathematical notation aside, the motivation behind integral transforms is easy to understand. There are many classes of problems that are difficult to solve—or at least quite unwieldy algebraically—in their original representations. An integral transform "maps" an equation from its original "domain" into another domain. Manipulating and solving the equation in the target domain can be much easier than manipulation and solution in the original domain.

In consequence, the Jacobian conjecture is true either for all fields of characteristic 0 or for none. For fixed N, it is true if it holds for at least one algebraically closed field of characteristic 0. Let k[X] denote the polynomial ring and k[F] denote the k-subalgebra generated by f1, ..., fn. For a given F, the Jacobian conjecture is true if, and only if, .

There is also an algebraic version of incompressibility. Suppose \iota: S \rightarrow M is a proper embedding of a compact surface in a 3-manifold. Then S is π1-injective (or algebraically incompressible) if the induced map :\iota_\star: \pi_1(S) \rightarrow \pi_1(M) on fundamental groups is injective. In general, every π1-injective surface is incompressible, but the reverse implication is not always true.

Claspers may also be interpreted algebraically, as a diagram calculus for the braided strict monoidal category Cob of oriented connected surfaces with connected boundary. Additionally, most crucially, claspers may be roughly viewed as a topological realization of Jacobi diagrams, which are purely combinatorial objects. This explains the Lie algebra structure of the graded vector space of Jacobi diagrams in terms of the Hopf algebra structure of Cob.

If every polynomial over F of prime degree has a root in F, then every non-constant polynomial has a root in F.Shipman, J. Improving the Fundamental Theorem of Algebra The Mathematical Intelligencer, Volume 29 (2007), Number 4. pp. 9-14 It follows that a field is algebraically closed if and only if every polynomial over F of prime degree has a root in F.

We work over the polynomial ring Fq[T] of one variable over a finite field Fq with q elements. The completion C∞ of an algebraic closure of the field Fq((T−1)) of formal Laurent series in T−1 will be useful. It is a complete and algebraically closed field. First we need analogues to the factorials, which appear in the definition of the usual exponential function.

A formally real field with no formally real proper algebraic extension is a real closed field.Rajwade (1993) p.216 If K is formally real and Ω is an algebraically closed field containing K, then there is a real closed subfield of Ω containing K. A real closed field can be ordered in a unique way, and the non-negative elements are exactly the squares.

A system is zero-dimensional if it has a finite number of complex solutions (or solutions in an algebraically closed field). This terminology comes from the fact that the algebraic variety of the solutions has dimension zero. A system with infinitely many solutions is said to be positive-dimensional. A zero-dimensional system with as many equations as variables is sometimes said to be well-behaved.

Nimber multiplication (nim-multiplication) is defined recursively by :. Except for the fact that nimbers form a proper class and not a set, the class of nimbers determines an algebraically closed field of characteristic 2. The nimber additive identity is the ordinal 0, and the nimber multiplicative identity is the ordinal 1. In keeping with the characteristic being 2, the nimber additive inverse of the ordinal is itself.

Hironaka's variety is a smooth 3-dimensional complete variety but is not projective as it has a non-trivial curve algebraically equivalent to 0. Any 2-dimensional smooth complete variety is projective, so 3 is the smallest possible dimension for such an example. There are plenty of 2-dimensional complex manifolds that are not algebraic, such as Hopf surfaces (non Kähler) and non-algebraic tori (Kähler).

The Sylvester–Gallai theorem has been generalized to colored point sets in the Euclidean plane, and to systems of points and lines defined algebraically or by distances in a metric space. In general, these variations of the theorem consider only finite sets of points, to avoid examples like the set of all points in the Euclidean plane, which does not have an ordinary line.

The main strategy of the algorithm is to take covariant derivatives of the Riemann tensor. Cartan showed that in n dimensions at most n(n+1)/2 differentiations suffice. If the Riemann tensor and its derivatives of the one manifold are algebraically compatible with the other, then the two manifolds are isometric. The Cartan–Karlhede algorithm therefore acts as a kind of generalization of the Petrov classification.

Lie–Butcher theory combines classical B-series with Lie- series. Algebraically, B-series are based on pre-Lie algebras, whereas the generalised Lie–Butcher series are based on the concept of post-Lie algebras. Applications in numerical integration on manifolds has led to fundamental research in the structure of post-Lie algebras and their relations to differential geometry, with ramifications in many different areas of applications.

Two basic types of false position method can be distinguished historically, simple false position and double false position. Simple false position is aimed at solving problems involving direct proportion. Such problems can be written algebraically in the form: determine such that :ax = b , if and are known. The method begins by using a test input value , and finding the corresponding output value by multiplication: .

By the definitions, an abelian variety is a group variety. Its group of points can be proven to be commutative. For C, and hence by the Lefschetz principle for every algebraically closed field of characteristic zero, the torsion group of an abelian variety of dimension g is isomorphic to (Q/Z)2g. Hence, its n-torsion part is isomorphic to (Z/nZ)2g, i.e.

Following on from the example, in algebraic geometry one studies algebraic sets, i.e. subsets of Kn (where K is an algebraically closed field) that are defined as the common zeros of a set of polynomials in n variables. If A is such an algebraic set, one considers the commutative ring R of all polynomial functions A → K. The maximal ideals of R correspond to the points of A (because K is algebraically closed), and the prime ideals of R correspond to the subvarieties of A (an algebraic set is called irreducible or a variety if it cannot be written as the union of two proper algebraic subsets). The spectrum of R therefore consists of the points of A together with elements for all subvarieties of A. The points of A are closed in the spectrum, while the elements corresponding to subvarieties have a closure consisting of all their points and subvarieties.

The poset of positive integers has deviation 0: every descending chain is finite, so the defining condition for deviation is vacuously true. However, its opposite poset has deviation 1. Let k be an algebraically closed field and consider the poset of ideals of the polynomial ring k[x] in one variable. Since the deviation of this poset is the Krull dimension of the ring, we know that it should be 1.

The expression for this impedance determines the response of the filter and vice versa, and a realisation of the filter can be obtained by expansion of this expression. It is not possible to realise any arbitrary impedance expression as a network. Foster's reactance theorem stipulates necessary and sufficient conditions for realisability: that the reactance must be algebraically increasing with frequency and the poles and zeroes must alternate.Cauer et al.

So taking any basis in these vector spaces we obtain functions Hi, which are Hitchin's hamiltonians. The construction for general reductive group is similar and uses invariant polynomials on the Lie algebra of G. For trivial reasons these functions are algebraically independent, and some calculations show that their number is exactly half of the dimension of the phase space. The nontrivial part is a proof of Poisson commutativity of these functions.

In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broadest of the "Bertini theorems" applying to a linear system of divisors; simplest because there is no restriction on the characteristic of the underlying field, while the extensions require characteristic 0.Hartshorne, Ch. III.10.

Thus, if, say, k is algebraically closed, then all p_i's are of the form t - \lambda_i and the above decomposition corresponds to the Jordan canonical form of f. In algebraic geometry, UFDs arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a regular local ring. A regular local ring is a UFD.

On the other hand, if F is not algebraically closed, then there is some non-constant polynomial p(x) in F[x] without roots in F. Let q(x) be some irreducible factor of p(x). Since p(x) has no roots in F, q(x) also has no roots in F. Therefore, q(x) has degree greater than one, since every first degree polynomial has one root in F.

Choose C and D so that P has an automorphism σ of order 2 acting freely on P and exchanging C and D, and also exchanging c and d. Then the quotient of V by the action of σ is a smooth 3-dimensional algebraic space with an irreducible curve algebraically equivalent to 0. This means that the quotient is a smooth 3-dimensional algebraic space that is not a scheme.

More generally, for a nonsingular algebraic curve C defined over an algebraically closed field k of characteristic p \geq 0, the gap numbers for all but finitely many points is a fixed sequence \epsilon_1, ..., \epsilon_g. These points are called non- Weierstrass points. All points of C whose gap sequence is different are called Weierstrass points. If \epsilon_1, ..., \epsilon_g = 1, ..., g then the curve is called a classical curve.

In fact, this is another way to state the Lie–Kolchin theorem. Lie's theorem states that any nonzero representation of a solvable Lie algebra on a finite dimensional vector space over an algebraically closed field of characteristic 0 has a one-dimensional invariant subspace. The result for Lie algebras was proved by and for algebraic groups was proved by . The Borel fixed point theorem generalizes the Lie–Kolchin theorem.

AIP, 2010.). Unfortunately, the closure (slaving relations) are algebraically unknown (as otherwise the coarse evolution law would be known). Initializing the unknown microscale modes randomly introduces a lifting error: we rely on the separation of macro and micro time scales to ensure a quick relaxation to functionals of the coarse macrostates (healing). A preparatory step may be required, possibly involving microscale simulations constrained to keep the macrostates fixed.

Three subfields of the complex numbers C have been suggested as encoding the notion of a "closed-form number"; in increasing order of generality, these are the Liouville numbers (not to be confused with Liouville numbers in the sense of rational approximation), EL numbers and elementary numbers. The Liouville numbers, denoted L, form the smallest algebraically closed subfield of C closed under exponentiation and logarithm (formally, intersection of all such subfields)—that is, numbers which involve explicit exponentiation and logarithms, but allow explicit and implicit polynomials (roots of polynomials); this is defined in . L was originally referred to as elementary numbers, but this term is now used more broadly to refer to numbers defined explicitly or implicitly in terms of algebraic operations, exponentials, and logarithms. A narrower definition proposed in , denoted E, and referred to as EL numbers, is the smallest subfield of C closed under exponentiation and logarithm—this need not be algebraically closed, and correspond to explicit algebraic, exponential, and logarithmic operations.

More precisely, there is an equivalence of categories between smooth proper algebraic curves over an algebraically closed field and finite field extensions of . In higher dimension the function field remembers less, but still decisive information about . The study of function fields and their geometric meaning in higher dimensions is referred to as birational geometry. The minimal model program attempts to identify the simplest (in a certain precise sense) algebraic varieties with a prescribed function field.

In mathematics, and especially differential and algebraic geometry, K-stability is an algebro-geometric stability condition, for complex manifolds and complex algebraic varieties. The notion of K-stability was first introduced by Gang Tian and reformulated more algebraically later by Simon Donaldson. The definition was inspired by a comparison to geometric invariant theory (GIT) stability. In the special case of Fano varieties, K-stability precisely characterises the existence of Kähler–Einstein metrics.

There are various proofs of this theorem, by either analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one real root. Because of this fact, theorems that hold for any algebraically closed field apply to C. For example, any non-empty complex square matrix has at least one (complex) eigenvalue.

In graph theory, a branch of mathematics, the (binary) cycle space of an undirected graph is the set of its even-degree subgraphs. This set of subgraphs can be described algebraically as a vector space over the two- element finite field. The dimension of this space is the circuit rank of the graph. The same space can also be described in terms from algebraic topology as the first homology group of the graph.

The Riemann–Zariski space of a curve over an algebraically closed field k with function field K is the same as the nonsingular projective model of it. It has one generic non- closed point corresponding to the trivial valuation with valuation ring K, and its other points are the rank 1 valuation rings in K containing k. Unlike the higher-dimensional cases, the Zariski–Riemann space of a curve is a scheme.

It is very hard to construct examples of Noetherian rings that are not universally catenary. The first example was found by , who found a 2-dimensional Noetherian local domain that is catenary but not universally catenary. Nagata's example is as follows. Choose a field k and a formal power series z=Σi>0aixi in the ring S of formal power series in x over k such that z and x are algebraically independent.

Consider linear operators on a finite-dimensional vector space over a field. An operator T is semisimple if every T-invariant subspace has a complementary T-invariant subspace (if the underlying field is algebraically closed, this is the same as the requirement that the operator be diagonalizable). An operator x is nilpotent if some power xm of it is the zero operator. An operator x is unipotent if x − 1 is nilpotent.

A square metre quadrat made of PVC pipe. Every unit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres (m2), square centimetres (cm2), square millimetres (mm2), square kilometres (km2), square feet (ft2), square yards (yd2), square miles (mi2), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units.

In mathematics, a paratopological group is a topological semigroup that is algebraically a group.Artur Hideyuki Tomita. On sequentially compact both- sides cancellative semigroups with sequentially continuous addition. In other words, it is a group G with a topology such that the group's product operation is a continuous function from G × G to G. This differs from the definition of a topological group in that the group inverse is not required to be continuous.

The latter asserts that for two projective varieties V and W in Pr over an algebraically closed field, the connectedness dimension of Z = V ∩ W (i.e., the minimal dimension of a closed subset T of Z that has to be removed from Z so that the complement Z \ T is disconnected) is bound by :c(Z) ≥ dim V + dim W − r − 1\. For example, Z is connected if dim V + dim W > r.

Subgroups between a Borel subgroup B and the ambient group G are called parabolic subgroups. Parabolic subgroups P are also characterized, among algebraic subgroups, by the condition that G/P is a complete variety. Working over algebraically closed fields, the Borel subgroups turn out to be the minimal parabolic subgroups in this sense. Thus B is a Borel subgroup when the homogeneous space G/B is a complete variety which is "as large as possible".

For X a complex algebraic K3 surface, \rho can be any integer between 1 and 20. In the complex analytic case, \rho may also be zero. (In that case, X contains no closed complex curves at all. By contrast, an algebraic surface always contains many continuous families of curves.) Over an algebraically closed field of characteristic p > 0, there is a special class of K3 surfaces, supersingular K3 surfaces, with Picard number 22.

The main initial step in calculating étale cohomology groups of a variety is to calculate them for complete connected smooth algebraic curves X over algebraically closed fields k. The étale cohomology groups of arbitrary varieties can then be controlled using analogues of the usual machinery of algebraic topology, such as the spectral sequence of a fibration. For curves the calculation takes several steps, as follows . Let Gm denote the sheaf of non-vanishing functions.

A vector manifold is always a subset of Universal Geometric Algebra by definition and the elements can be manipulated algebraically. In contrast, a manifold is not a subset of any set other than itself, but the elements have no algebraic relation among them. The differential geometry of a manifold can be carried out in a vector manifold. All quantities relevant to differential geometry can be calculated from if it is a differentiable function.

Replacing vectors by p-vectors (pth exterior power of vectors) yields p-vector fields; taking the dual space and exterior powers yields differential k-forms, and combining these yields general tensor fields. Algebraically, vector fields can be characterized as derivations of the algebra of smooth functions on the manifold, which leads to defining a vector field on a commutative algebra as a derivation on the algebra, which is developed in the theory of differential calculus over commutative algebras.

The quotient ring is a field, because is irreducible over , so the ideal it generates is maximal. The formulas for addition and multiplication in the ring , modulo the relation , correspond to the formulas for addition and multiplication of complex numbers defined as ordered pairs. So the two definitions of the field are isomorphic (as fields). Accepting that is algebraically closed, since it is an algebraic extension of in this approach, is therefore the algebraic closure of .

For each line or plane of reflection, the symmetry group is isomorphic with Cs (see point groups in three dimensions), one of the three types of order two (involutions), hence algebraically C2. The fundamental domain is a half-plane or half-space. In certain contexts there is rotational as well as reflection symmetry. Then mirror-image symmetry is equivalent to inversion symmetry; in such contexts in modern physics the term parity or P-symmetry is used for both.

Now any common model with an embedding from these two extensions must be at least of size five so that there are two elements on either side of e. Now consider the class of algebraically closed fields. This class has the amalgamation property since any two field extensions of a prime field can be embedded into a common field. However, two arbitrary fields cannot be embedded into a common field when the characteristic of the fields differ.

To prove the result Lang took two algebraically independent functions from , say, and , and then created an auxiliary function . Using Siegel's lemma, he then showed that one could assume vanished to a high order at the . Thus a high-order derivative of takes a value of small size at one such s, "size" here referring to an algebraic property of a number. Using the maximum modulus principle, Lang also found a separate estimate for absolute values of derivatives of .

Diophantine geometry, which is the application of techniques from algebraic geometry in this field, has continued to grow as a result; since treating arbitrary equations is a dead end, attention turns to equations that also have a geometric meaning. The central idea of Diophantine geometry is that of a rational point, namely a solution to a polynomial equation or a system of polynomial equations, which is a vector in a prescribed field , when is not algebraically closed.

For example, a circle and a line have always two intersection points (possibly not distinct) in the complex affine space. Therefore, most of algebraic geometry is built in complex affine spaces and affine spaces over algebraically closed fields. The shapes that are studied in algebraic geometry in these affine spaces are therefore called affine algebraic varieties. Affine spaces over the rational numbers and more generally over algebraic number fields provide a link between (algebraic) geometry and number theory.

A system is overdetermined if the number of equations is higher than the number of variables. A system is inconsistent if it has no complex solution (or, if the coefficients are not complex numbers, no solution in an algebraically closed field containing the coefficients). By Hilbert's Nullstellensatz this means that 1 is a linear combination (with polynomials as coefficients) of the first members of the equations. Most but not all overdetermined systems, when constructed with random coefficients, are inconsistent.

If N is a Galois extension of a Hilbertian field, then although N need not be Hilbertian itself, Weisseauer's results asserts that any proper finite extension of N is Hilbertian. The most general result in this direction is Haran's diamond theorem. A discussion on these results and more appears in Fried-Jarden's Field Arithmetic. Being Hilbertian is at the other end of the scale from being algebraically closed: the complex numbers have all sets thin, for example.

More generally, for any two transcendental numbers a and b, at least one of a + b and ab must be transcendental. To see this, consider the polynomial (x − a)(x − b) = x2 − (a + b)x + ab. If (a + b) and ab were both algebraic, then this would be a polynomial with algebraic coefficients. Because algebraic numbers form an algebraically closed field, this would imply that the roots of the polynomial, a and b, must be algebraic.

The Weyl tensor is algebraically special, in fact it has Petrov type D. The global structure is known. Topologically, the homotopy type of the Kerr spacetime can be simply characterized as a line with circles attached at each integer point. Note that the inner Kerr geometry is unstable with regard to perturbations in the interior region. This instability means that although the Kerr metric is axis-symmetric, a black hole created through gravitational collapse may not be so.

Two linear systems using the same set of variables are equivalent if each of the equations in the second system can be derived algebraically from the equations in the first system, and vice versa. Two systems are equivalent if either both are inconsistent or each equation of each of them is a linear combination of the equations of the other one. It follows that two linear systems are equivalent if and only if they have the same solution set.

In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field.... For example, x^2+y^2-1 is absolutely irreducible, but while x^2+y^2 is irreducible over the integers and the reals, it is reducible over the complex numbers as x^2+y^2 = (x+iy)(x-iy), and thus not absolutely irreducible. More generally, a polynomial defined over a field K is absolutely irreducible if it is irreducible over every algebraic extension of K,. and an affine algebraic set defined by equations with coefficients in a field K is absolutely irreducible if it is not the union of two algebraic sets defined by equations in an algebraically closed extension of K. In other words, an absolutely irreducible algebraic set is a synonym of an algebraic variety,. which emphasizes that the coefficients of the defining equations may not belong to an algebraically closed field. Absolutely irreducible is also applied, with the same meaning to linear representations of algebraic groups.

We can thus substitute for: Saving + Trade Deficit = Investment + Budget Deficit. Rearranging algebraically we find that: Budget Deficit = Saving + Trade Deficit – Investment. What we can gather from this is the understanding of why an increased budget deficit goes up and down in tandem with the Trade Deficit. This is where we derive the appellation the Twin Deficits: if the US budget deficit goes up then either household savings must go up, the trade deficit must go up, or private investment will decrease.

Abstract mathematical problems arise in all fields of mathematics. While mathematicians usually study them for their own sake, by doing so results may be obtained that find application outside the realm of mathematics. Theoretical physics has historically been, and remains, a rich source of inspiration. Some abstract problems have been rigorously proved to be unsolvable, such as squaring the circle and trisecting the angle using only the compass and straightedge constructions of classical geometry, and solving the general quintic equation algebraically.

Grassmann's laws describe empirical results about how the perception of mixtures of colored lights (i.e., lights that co-stimulate the same area on the retina) composed of different spectral power distributions can be algebraically related to one another in a color matching context. Discovered by Hermann Grassmann these "laws" are actually principles used to predict color match responses to a good approximation under photopic and mesopic vision. A number of studies have examined how and why they provide poor predictions under specific conditions.

Debt sculpting is common in the financial modelling of a project. It means that the principal repayment obligations have been calculated to ensure that the principal and interest obligations are appropriately matched to the strength and pattern of the cashflows in each period. The most common ways to do so are to manually adjust the principal repayment in each period, or to algebraically solve the principal repayment to achieve a desired DSCR. See Cashflow matching, Immunization (finance), Asset–liability mismatch.

Lee, J. M., Introduction to Topological Manifolds, Springer 2011, , p153 For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1). In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems. An abstract simplex can be studied algebraically by forming its Stanley–Reisner ring; this sets up a powerful relation between combinatorics and commutative algebra.

For avoiding ambiguity, the term irreducible hypersurface is often used. As for algebraic varieties, the coefficients of the defining polynomial may belong to any fixed field , and the points of the hypersurface are the zeros of in the affine space K^n, where is an algebraically closed extension of . A hypersurface may have singularities, which are the common zeros, if any, of the defining polynomial and its partial derivatives. In particular, a real algebraic hypersurface is not necessarily a manifold.

The rigidity matroid of the given framework is a linear matroid that has as its elements the edges of the graph. A set of edges is independent, in the matroid, if it corresponds to a set of rows of the rigidity matrix that is linearly independent. A framework is called generic if the coordinates of its vertices are algebraically independent real numbers. Any two generic frameworks on the same graph G determine the same rigidity matroid, regardless of their specific coordinates.

The word problem for a finitely generated group is the decision problem whether two words in the generators of the group represent the same element. The word problem for a given finitely generated group is solvable if and only if the group can be embedded in every algebraically closed group. The rank of a group is often defined to be the smallest cardinality of a generating set for the group. By definition, the rank of a finitely generated group is finite.

The general treatment of nuisance parameters can be broadly similar between frequentist and Bayesian approaches to theoretical statistics. It relies on an attempt to partition the likelihood function into components representing information about the parameters of interest and information about the other (nuisance) parameters. This can involve ideas about sufficient statistics and ancillary statistics. When this partition can be achieved it may be possible to complete a Bayesian analysis for the parameters of interest by determining their joint posterior distribution algebraically.

In other words, increasing the exponent up to will give ever larger kernels, but further increasing the exponent beyond will just give the same kernel. If the field is not algebraically closed, then the minimal and characteristic polynomials need not factor according to their roots (in ) alone, in other words they may have irreducible polynomial factors of degree greater than . For irreducible polynomials one has similar equivalences: # divides , # divides , # the kernel of has dimension at least . # the kernel of has dimension at least .

The field F is algebraically closed if and only if it has no proper algebraic extension. If F has no proper algebraic extension, let p(x) be some irreducible polynomial in F[x]. Then the quotient of F[x] modulo the ideal generated by p(x) is an algebraic extension of F whose degree is equal to the degree of p(x). Since it is not a proper extension, its degree is 1 and therefore the degree of p(x) is 1\.

In mathematics, an Artin–Schreier curve is a plane curve defined over an algebraically closed field of characteristic p by an equation :y^p - y = f(x) for some rational function f over that field. One of the most important examples of such curves is hyperelliptic curves in characteristic 2, whose Jacobian varieties have been suggested for use in cryptography. It is common to write these curves in the form :y^2 + h(x) y = f(x) for some polynomials f and h.

The main property of linear underdetermined systems, of having either no solution or infinitely many, extends to systems of polynomial equations in the following way. A system of polynomial equations which has fewer equations than unknowns is said to be underdetermined. It has either infinitely many complex solutions (or, more generally, solutions in an algebraically closed field) or is inconsistent. It is inconsistent if and only if is a linear combination (with polynomial coefficients) of the equations (this is Hilbert's Nullstellensatz).

Within a system of classical logic, double negation, that is, the negation of the negation of a proposition P, is logically equivalent to P. Expressed in symbolic terms, eg eg P \equiv P. In intuitionistic logic, a proposition implies its double negation, but not conversely. This marks one important difference between classical and intuitionistic negation. Algebraically, classical negation is called an involution of period two. However, in intuitionistic logic, the equivalence eg eg eg P \equiv eg P does not hold.

The symmetries of the Dynkin diagram, , correspond to the outer automorphisms of in triality. Let now be a connected reductive group over an algebraically closed field. Then any two Borel subgroups are conjugate by an inner automorphism, so to study outer automorphisms it suffices to consider automorphisms that fix a given Borel subgroup. Associated to the Borel subgroup is a set of simple roots, and the outer automorphism may permute them, while preserving the structure of the associated Dynkin diagram.

A pseudo algebraically closed field (in short PAC) K is a field satisfying the following geometric property. Each absolutely irreducible algebraic variety V defined over K has a K-rational point. Over PAC fields there is a firm link between arithmetic properties of the field and group theoretic properties of its absolute Galois group. A nice theorem in this spirit connects Hilbertian fields with ω-free fields (K is ω-free if any embedding problem for K is properly solvable). Theorem.

The simple group SO(q) can always be defined as the maximal smooth connected subgroup of O(q) over k.) When k is algebraically closed, any two (nondegenerate) quadratic forms of the same dimension are isomorphic, and so it is reasonable to call this group SO(n). For a general field k, different quadratic forms of dimension n can yield non- isomorphic simple groups SO(q) over k, although they all have the same base change to the algebraic closure \overline k.

The theory of algebraically closed fields of a given characteristic is complete, consistent, and has an infinite but recursively enumerable set of axioms. However it is not possible to encode the integers into this theory, and the theory cannot describe arithmetic of integers. A similar example is the theory of real closed fields, which is essentially equivalent to Tarski's axioms for Euclidean geometry. So Euclidean geometry itself (in Tarski's formulation) is an example of a complete, consistent, effectively axiomatized theory.

Noether's normalisation lemma is a theorem in commutative algebra. Given a field K and a finitely generated K-algebra A, the theorem says it is possible to find elements y1, y2, ..., ym in A that are algebraically independent over K such that A is finite (and hence integral) over B = K[y1,..., ym]. Thus the extension K ⊂ A can be written as a composite K ⊂ B ⊂ A where K ⊂ B is a purely transcendental extension and B ⊂ A is finite.Chapter 4 of Reid.

Reductive groups include the most important linear algebraic groups in practice, such as the classical groups: GL(n), SL(n), the orthogonal groups SO(n) and the symplectic groups Sp(2n). On the other hand, the definition of reductive groups is quite "negative", and it is not clear that one can expect to say much about them. Remarkably, Claude Chevalley gave a complete classification of the reductive groups over an algebraically closed field: they are determined by root data.Springer (1998), 9.6.

In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algebraically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite; this definition is more restrictive and excludes tori,. A compact Lie algebra can be seen as the smallest real form of a corresponding complex Lie algebra, namely the complexification.

An affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define projective and quasi-projective varieties in a similar way. The most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, but Nagata gave an example of such a new variety in the 1950s.

One way that leads to generalizations is to allow reducible algebraic sets (and fields that aren't algebraically closed), so the rings R may not be integral domains. A more significant modification is to allow nilpotents in the sheaf of rings, that is, rings which are not reduced. This is one of several generalizations of classical algebraic geometry that are built into Grothendieck's theory of schemes. Allowing nilpotent elements in rings is related to keeping track of "multiplicities" in algebraic geometry.

Eduard Weyr Eduard Weyr (June 22, 1852 – July 23, 1903) was a Czech mathematician now chiefly remembered as the discoverer of a certain canonical form for square matrices over algebraically closed fields. Weyr presented this form briefly in a paper published in 1885. He followed it up with a more elaborate treatment in a paper published in 1890. This particular canonical form has been named as the Weyr canonical form in a paper by Shapiro published in The American Mathematical Monthly in 1999.

An order theoretic meet-semilattice gives rise to a binary operation ∧ such that is an algebraic meet-semilattice. Conversely, the meet-semilattice gives rise to a binary relation ≤ that partially orders S in the following way: for all elements x and y in S, x ≤ y if and only if x = x ∧ y. The relation ≤ introduced in this way defines a partial ordering from which the binary operation ∧ may be recovered. Conversely, the order induced by the algebraically defined semilattice coincides with that induced by ≤.

A field containing is called an algebraic closure of if it is algebraic over (roughly speaking, not too big compared to ) and is algebraically closed (big enough to contain solutions of all polynomial equations). By the above, is an algebraic closure of . The situation that the algebraic closure is a finite extension of the field is quite special: by the Artin-Schreier theorem, the degree of this extension is necessarily 2, and is elementarily equivalent to . Such fields are also known as real closed fields.

An affine quadric is the set of zeros of a polynomial of degree two. When not specified otherwise, the polynomial is supposed to have real coefficients, and the zeros are points in a Euclidean space. However, most properties remain true when the coefficients belong to any field and the points belong in an affine space. As usually in algebraic geometry, it is often useful to consider points over an algebraically closed field containing the polynomial coefficients, generally the complex numbers, when the coefficients are real.

Thus a = b. But this may not be considered direct as one must first argue about why the other factor cannot be 0\. John ConwayAlleged impossibility of "direct" proof of Steiner–Lehmus theorem has argued that there can be no "equality-chasing" proof because the theorem (stated algebraically) does not hold over an arbitrary field, or even when negative real numbers are allowed as parameters. A precise definition of a "direct proof" inside both classical and intuitionistic logic has been provided by Victor Pambuccian.

The field of real numbers is not algebraically closed, the geometry of even a plane curve C in the real projective plane. Assuming no singular points, the real points of C form a number of ovals, in other words submanifolds that are topologically circles. The real projective plane has a fundamental group that is a cyclic group with two elements. Such an oval may represent either group element; in other words we may or may not be able to contract it down in the plane.

Polynomial equations of degree up to four can be solved exactly using algebraic methods, of which the quadratic formula is the simplest example. Polynomial equations with a degree of five or higher require in general numerical methods (see below) or special functions such as Bring radicals, although some specific cases may be solvable algebraically, for example :4x^5 - x^3 - 3 = 0 (by using the rational root theorem), and :x^6 - 5x^3 + 6 = 0 \, , (by using the substitution , which simplifies this to a quadratic equation in ).

In mathematics, a pseudo-finite field F is an infinite model of the first- order theory of finite fields. This is equivalent to the condition that F is quasi-finite (perfect with a unique extension of every positive degree) and pseudo algebraically closed (every absolutely irreducible variety over F has a point defined over F). Every hyperfinite field is pseudo-finite and every pseudo-finite field is quasifinite. Every non-principal ultraproduct of finite fields is pseudo-finite. Pseudo-finite fields were introduced by .

Hilbert's Nullstellensatz suggests an approach to algebraic geometry over any algebraically closed field k: the maximal ideals in the polynomial ring k[x1,...,xn] are in one-to-one correspondence with the set kn of n-tuples of elements of k, and the prime ideals correspond to the irreducible algebraic sets in kn, known as affine varieties. Motivated by these ideas, Emmy Noether and Wolfgang Krull developed the subject of commutative algebra in the 1920s and 1930s.Dieudonné (1985), sections VII.2 and VII.5.

The fundamental theorem of algebra states that every non-constant single- variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots.

For example, the system is overdetermined (having two equations but only one unknown), but it is not inconsistent since it has the solution . A system is underdetermined if the number of equations is lower than the number of the variables. An underdetermined system is either inconsistent or has infinitely many complex solutions (or solutions in an algebraically closed field that contains the coefficients of the equations). This is a non-trivial result of commutative algebra that involves, in particular, Hilbert's Nullstellensatz and Krull's principal ideal theorem.

There is also a logical facet to elimination theory, as seen in the Boolean satisfiability problem. In the worst case, it is presumably hard to eliminate variables computationally. Quantifier elimination is a term used in mathematical logic to explain that, in some theories, every formula is equivalent to a formula without quantifier. This is the case of the theory of polynomials over an algebraically closed field, where elimination theory may be viewed as the theory of the methods to make quantifier elimination algorithmically effective.

However, it is more common and simpler algebraically to use coordinates where the equation of the line is lx + my + 1 = 0\. This system specifies coordinates for all lines except those that pass through the origin. The geometrical interpretations of l and m are the negative reciprocals of the x and y-intercept respectively. The exclusion of lines passing through the origin can be resolved by using a system of three coordinates to specify the line in which the equation, lx + my + n = 0\.

The first chapter, titled "Varieties", deals with the classical algebraic geometry of varieties over algebraically closed fields. This chapter uses many classical results in commutative algebra, including Hilbert's Nullstellensatz, with the books by Atiyah-Macdonald, Matsumura, and Zariski-Samuel as usual references. The second and the third chapters, "Schemes" and "Cohomology", form the technical heart of the book. The last two chapters, "Curves" and "Surfaces", respectively explore the geometry of 1- and 2-dimensional objects, using the tools developed in the chapters 2 and 3.

Introduction to commutative algebra. Addison-Wesley publishing Company. pp. 11–12.Kaplansky (1972) pp.74-76 or the weaker ultrafilter lemma,Mathoverflow discussion it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K. The algebraic closure of a field K can be thought of as the largest algebraic extension of K. To see this, note that if L is any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is contained within the algebraic closure of K. The algebraic closure of K is also the smallest algebraically closed field containing K, because if M is any algebraically closed field containing K, then the elements of M that are algebraic over K form an algebraic closure of K. The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite.

By means of a Chevalley basis for the Lie algebra, one can define E8 as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimes also known as “untwisted”) form of E8. Over an algebraically closed field, this is the only form; however, over other fields, there are often many other forms, or “twists” of E8, which are classified in the general framework of Galois cohomology (over a perfect field k) by the set H1(k,Aut(E8)) which, because the Dynkin diagram of E8 (see below) has no automorphisms, coincides with H1(k,E8). (English translation: ), §2.2.4 Over R, the real connected component of the identity of these algebraically twisted forms of E8 coincide with the three real Lie groups mentioned above, but with a subtlety concerning the fundamental group: all forms of E8 are simply connected in the sense of algebraic geometry, meaning that they admit no non-trivial algebraic coverings; the non-compact and simply connected real Lie group forms of E8 are therefore not algebraic and admit no faithful finite-dimensional representations.

The Steiner–Lehmus theorem can be proved using elementary geometry by proving the contrapositive statement. There is some controversy over whether a "direct" proof is possible; allegedly "direct" proofs have been published, but not everyone agrees that these proofs are "direct." For example, there exist simple algebraic expressions for angle bisectors in terms of the sides of the triangle. Equating two of these expressions and algebraically manipulating the equation results in a product of two factors which equal 0, but only one of them (a − b) can equal 0 and the other must be positive.

Second, if V is an irreducible representation of a Lie group or Lie algebra over an algebraically closed field and \phi:V\rightarrow V is an intertwining map, then \phi is a scalar multiple of the identity map. Third, suppose V_1 and V_2 are irreducible representations of a Lie group or Lie algebra and \phi_1, \phi_2:V_1\rightarrow V_2 are nonzero intertwining maps. Then \phi_1=\lambda\phi_2 for some scalar \lambda. As a simple corollary of the second statement is that every complex irreducible representation of an Abelian group is one-dimensional.

Notably, the operations of addition and multiplication take on a very natural geometric character, when complex numbers are viewed as position vectors: addition corresponds to vector addition, while multiplication (see below) corresponds to multiplying their magnitudes and adding the angles they make with the real axis. Viewed in this way, the multiplication of a complex number by corresponds to rotating the position vector counterclockwise by a quarter turn (90°) about the origin—a fact which can be expressed algebraically as follows: : (a + bi)\cdot i = ai + b(i)^2 = -b + ai .

In mathematics, an absolute presentation is one method of defining a group.B. Neumann, The isomorphism problem for algebraically closed groups, in: Word Problems, Decision Problems, and the Burnside Problem in Group Theory, Amsterdam-London (1973), pp. 553–562. Recall that to define a group G\ by means of a presentation, one specifies a set S\ of generators so that every element of the group can be written as a product of some of these generators, and a set R\ of relations among those generators. In symbols: :G \simeq \langle S \mid R \rangle.

In 1861, Blissard published "Theory of generic equations" describing a system that allows certain sequences of numbers, , , , …, to be manipulated algebraically as powers, , , , …, so as to produce valid formulas. The Bernoulli numbers are a prime example of such a sequence. Blissard called his system "representative notation"; it is often referred to as "Blissard's symbolic method", but is most often called "umbral calculus", a term coined by J. J. Sylvester. Blissard published nine additional papers in The Quarterly Journal of Pure and Applied Mathematics, many of them applying the representative notation and developing it further.

Strictly speaking, the theory of the additive natural numbers did not admit quantifier elimination, but it was an expansion of the additive natural numbers that was shown to be decidable. Whenever a theory is decidable, and the language of its valid formulas is countable, it is possible to extend the theory with countably many relations to have quantifier elimination (for example, one can introduce, for each formula of the theory, a relation symbol that relates the free variables of the formula). Example: Nullstellensatz for algebraically closed fields and for differentially closed fields.

In algebraic geometry, a generic point of an algebraic variety is a point whose coordinates do not satisfy any other algebraic relation than those satisfied by every point of the variety. For example, a generic point of an affine space over a field is a point whose coordinates are algebraically independent over . In scheme theory, where the points are the sub varieties, a generic point of a variety is a point whose closure for the Zariski topology is the whole variety. A generic property is a property of the generic point.

Temporal summation occurs when a high frequency of action potentials in the presynaptic neuron elicits postsynaptic potentials that summate with each other. The duration of a postsynaptic potential is longer than the interval between action potentials. If the time constant of the cell membrane is sufficiently long, as is the case for the cell body, then the amount of summation is increased. The amplitude of one postsynaptic potential at the time point when the next one begins will algebraically summate with it, generating a larger potential than the individual potentials.

The local ring Rm is a regular local ring of dimension 1 (the proof of this uses the fact that z and x are algebraically independent) and the local ring Rn is a regular Noetherian local ring of dimension 2. Let B be the localization of R with respect to all elements not in either m or n. Then B is a 2-dimensional Noetherian semi-local ring with 2 maximal ideals, mB (of height 1) and nB (of height 2). Let I be the Jacobson radical of B, and let A = k+I.

It is often useful for the sessile drop technique to use liquids that are known to be incapable of some of those interactions (see table 1). For example, the surface tension of all straight alkanes is said to be entirely dispersive, and all of the other components are zero. This is algebraically useful, as it eliminates a variable in certain cases, and makes these liquids essential testing materials. The overall surface energy, both for a solid and a liquid, is assumed traditionally to simply be the sum of the components considered.

Solutions to wave propagation problems in linear elastic transversely isotropic media can be constructed by superposing solutions for the quasi-P wave, the quasi S-wave, and a S-wave polarized orthogonal to the quasi S-wave. However, the equations for the angular variation of velocity are algebraically complex and the plane-wave velocities are functions of the propagation angle \theta are. The direction dependent wave speeds for elastic waves through the material can be found by using the Christoffel equation and are given byG. Mavko, T. Mukerji, J. Dvorkin.

In mathematics, and especially differential topology and singularity theory, the Eisenbud–Levine–Khimshiashvili signature formula gives a way of computing the Poincaré–Hopf index of a real, analytic vector field at an algebraically isolated singularity. It is named after David Eisenbud, Harold I. Levine, and George Khimshiashvili. Intuitively, the index of a vector field near a zero is the number of times the vector field wraps around the sphere. Because analytic vector fields have a rich algebraic structure, the techniques of commutative algebra can be brought to bear to compute their index.

In number theory, a Pillai prime is a prime number p for which there is an integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a multiple of n. To put it algebraically, n! \equiv -1 \mod p but p ot\equiv 1 \mod n. The first few Pillai primes are :23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, ... Pillai primes are named after the mathematician Subbayya Sivasankaranarayana Pillai, who studied these numbers.

József Kürschák (14 March 1864 – 26 March 1933) was a Hungarian mathematician noted for his work on trigonometry and for his creation of the theory of valuations. He proved that every valued field can be embedded into a complete valued field which is algebraically closed. In 1918 he proved that the sum of reciprocals of consecutive natural numbers is never an integer. Extending Hilbert's argument, he proved that everything that can be constructed using a ruler and a compass, can be constructed by using a ruler and the ability of copying a fixed segment.

However, many arguments in algebraic geometry work better for projective varieties, essentially because projective varieties are compact. From the 1920s to the 1940s, B. L. van der Waerden, André Weil and Oscar Zariski applied commutative algebra as a new foundation for algebraic geometry in the richer setting of projective (or quasi-projective) varieties.Dieudonné (1985), section VII.4. In particular, the Zariski topology is a useful topology on a variety over any algebraically closed field, replacing to some extent the classical topology on a complex variety (based on the topology of the complex numbers).

Courtois and Pieprzyk (2002) observed that AES (Rijndael) and partially also Serpent could be expressed as a system of quadratic equations. The variables represent not just the plaintext, ciphertext and key bits, but also various intermediate values within the algorithm. The S-box of AES appears to be especially vulnerable to this type of analysis, as it is based on the algebraically simple inverse function. Subsequently, other ciphers have been studied to see what systems of equations can be produced (Biryukov and De Cannière, 2003), including Camellia, KHAZAD, MISTY1 and KASUMI.

An order-theoretic lattice gives rise to the two binary operations ∨ and ∧. Since the commutative, associative and absorption laws can easily be verified for these operations, they make into a lattice in the algebraic sense. The converse is also true. Given an algebraically defined lattice , one can define a partial order ≤ on L by setting : if , or : if , for all elements a and b from L. The laws of absorption ensure that both definitions are equivalent: a = a ∧ b implies b = b ∨ (b ∧ a) = (a ∧ b) ∨ b = a ∨ b and dually for the other direction.

In mathematics, the Ax–Grothendieck theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and Alexander Grothendieck... The theorem is often given as this special case: If P is an injective polynomial function from an n-dimensional complex vector space to itself then P is bijective. That is, if P always maps distinct arguments to distinct values, then the values of P cover all of Cn. The full theorem generalizes to any algebraic variety over an algebraically closed field.Éléments de géométrie algébrique, IV3, Proposition 10.4.11.

Other fields of coefficients, such as the real numbers, are less often used, as their elements cannot be represented in a computer (only approximations of real numbers can be used in computations, and these approximations are always rational numbers). A solution of a polynomial system is a tuple of values of that satisfies all equations of the polynomial system. The solutions are sought in the complex numbers, or more generally in an algebraically closed field containing the coefficients. In particular, in characteristic zero, all complex solutions are sought.

This became Hamilton's rule: in each behaviour-evoking situation, the individual assesses his neighbour's fitness against his own according to the coefficients of relationship appropriate to the situation. Algebraically, the rule posits that a costly action should be performed if: C < r \times B Where C is the cost in fitness to the actor, r the genetic relatedness between the actor and the recipient, and B is the fitness benefit to the recipient. Fitness costs and benefits are measured in fecundity. r is a number between 0 and 1.

In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group GLn (n x n invertible matrices), the subgroup of invertible upper triangular matrices is a Borel subgroup. For groups realized over algebraically closed fields, there is a single conjugacy class of Borel subgroups. Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally, reductive) algebraic groups, in Jacques Tits' theory of groups with a (B,N) pair.

Then and I form an antitone Galois connection. The closure on is the closure in the Zariski topology, and if the field is algebraically closed, then the closure on the polynomial ring is the radical of ideal generated by . More generally, given a commutative ring (not necessarily a polynomial ring), there is an antitone Galois connection between radical ideals in the ring and subvarieties of the affine variety . More generally, there is an antitone Galois connection between ideals in the ring and subschemes of the corresponding affine variety.

The dot product may be defined algebraically or geometrically. The geometric definition is based on the notions of angle and distance (magnitude of vectors). The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space. In modern presentations of Euclidean geometry, the points of space are defined in terms of their Cartesian coordinates, and Euclidean space itself is commonly identified with the real coordinate space Rn. In such a presentation, the notions of length and angles are defined by means of the dot product.

3 selections with decimal odds a, b and c. Expanding (a + 1)(b + 1)(c + 1) algebraically gives abc + ab + ac + bc + a + b + c + 1. This is equivalent to the OM for a Patent (treble: abc; doubles: ab, ac and bc; singles: a, b and c) plus 1. Therefore to calculate the returns for a winning Patent it is just a case of multiplying (a + 1), (b + 1) and (c + 1) together and subtracting 1 to get the OM for the winning bet, i.e. OM = (a + 1)(b + 1)(c + 1) − 1\.

Many sets of linear operators in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function. Any local field has a topology native to it, and this can be extended to vector spaces over that field. Every manifold has a natural topology since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from Rn. The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety.

Conjugate, or sequent, depths are the paired depths that result upstream and downstream of a hydraulic jump, with the upstream flow being supercritical and downstream flow being subcritical. Conjugate depths can be found either graphically using a specific momentum curve or algebraically with a set of equations. Because momentum is conserved over a hydraulic jump conjugate depths have equivalent momentum, and given a discharge, the conjugate to any flow depth can be determined with an M-y diagram (Figure 6). A vertical line that crosses the M-y curve twice (i.e.

Chang's (1958, 1959) completeness theorem states that any MV-algebra equation holding in the standard MV-algebra over the interval [0,1] will hold in every MV-algebra. Algebraically, this means that the standard MV-algebra generates the variety of all MV-algebras. Equivalently, Chang's completeness theorem says that MV- algebras characterize infinite-valued Łukasiewicz logic, defined as the set of [0,1]-tautologies. The way the [0,1] MV-algebra characterizes all possible MV- algebras parallels the well-known fact that identities holding in the two- element Boolean algebra hold in all possible Boolean algebras.

If is finite- dimensional, this can be rephrased using determinants: having eigenvalue is equivalent to :. By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in , called the characteristic polynomial of . If the field is large enough to contain a zero of this polynomial (which automatically happens for algebraically closed, such as ) any linear map has at least one eigenvector. The vector space may or may not possess an eigenbasis, a basis consisting of eigenvectors.

William Neile (7 December 1637 – 24 August 1670) was an English mathematician and founder member of the Royal Society. His major mathematical work, the rectification of the semicubical parabola, was carried out when he was aged nineteen, and was published by John Wallis. By carrying out the determination of arc lengths on a curve given algebraically, in other words by extending to algebraic curves generally with Cartesian geometry a basic concept from differential geometry, it represented a major advance in what would become infinitesimal calculus. His name also appears as Neil.

Table 2 from Noether's dissertation on invariant theory. This table collects 202 of the 331 invariants of ternary biquadratic forms. These forms are graded in two variables x and u. The horizontal direction of the table lists the invariants with increasing grades in x, while the vertical direction lists them with increasing grades in u. The ring of invariants is generated by 7 algebraically independent invariants of degrees 3, 6, 9, 12, 15, 18, 27 (discriminant) , together with 6 more invariants of degrees 9, 12, 15, 18, 21, 21, as conjectured by .

Mason's gain formula (MGF) is a method for finding the transfer function of a linear signal-flow graph (SFG). The formula was derived by Samuel Jefferson Mason, whom it is also named after. MGF is an alternate method to finding the transfer function algebraically by labeling each signal, writing down the equation for how that signal depends on other signals, and then solving the multiple equations for the output signal in terms of the input signal. MGF provides a step by step method to obtain the transfer function from a SFG.

Vieta's formulas are frequently used with polynomials with coefficients in any integral domain . Then, the quotients a_i/a_n belong to the ring of fractions of (and possibly are in itself if a_n happens to be invertible in ) and the roots r_i are taken in an algebraically closed extension. Typically, is the ring of the integers, the field of fractions is the field of the rational numbers and the algebraically closed field is the field of the complex numbers. Vieta's formulas are then useful because they provide relations between the roots without having to compute them. For polynomials over a commutative ring which is not an integral domain, Vieta's formulas are only valid when a_n is a non zero-divisor and P(x) factors as a_n(x-r_1)(x-r_2)\dots(x-r_n). For example, in the ring of the integers modulo 8, the polynomial P(x)=x^2-1 has four roots: 1, 3, 5, and 7. Vieta's formulas are not true if, say, r_1=1 and r_2=3, because P(x) eq (x-1)(x-3). However, P(x) does factor as (x-1)(x-7) and as (x-3)(x-5), and Vieta's formulas hold if we set either r_1=1 and r_2=7 or r_1=3 and r_2=5.

By means of a Chevalley basis for the Lie algebra, one can define E7 as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimes also known as “untwisted”) adjoint form of E7. Over an algebraically closed field, this and its double cover are the only forms; however, over other fields, there are often many other forms, or “twists” of E7, which are classified in the general framework of Galois cohomology (over a perfect field k) by the set H1(k, Aut(E7)) which, because the Dynkin diagram of E7 (see below) has no automorphisms, coincides with H1(k, E7, ad). (original version: ), §2.2.4 Over the field of real numbers, the real component of the identity of these algebraically twisted forms of E7 coincide with the three real Lie groups mentioned above, but with a subtlety concerning the fundamental group: all adjoint forms of E7 have fundamental group Z/2Z in the sense of algebraic geometry, meaning that they admit exactly one double cover; the further non-compact real Lie group forms of E7 are therefore not algebraic and admit no faithful finite-dimensional representations.

The method of quantifier elimination can be used to show that definable sets in particular theories cannot be too complicated. Tarski (1948) established quantifier elimination for real-closed fields, a result which also shows the theory of the field of real numbers is decidable. (He also noted that his methods were equally applicable to algebraically closed fields of arbitrary characteristic.) A modern subfield developing from this is concerned with o-minimal structures. Morley's categoricity theorem, proved by Michael D. Morley (1965), states that if a first-order theory in a countable language is categorical in some uncountable cardinality, i.e.

For fields that are not algebraically closed (or not separably closed), the absolute Galois group is fundamentally important: extending the case of finite Galois extensions outlined above, this group governs all finite separable extensions of . By elementary means, the group can be shown to be the Prüfer group, the profinite completion of . This statement subsumes the fact that the only algebraic extensions of are the fields for , and that the Galois groups of these finite extensions are given by :. A description in terms of generators and relations is also known for the Galois groups of -adic number fields (finite extensions of ).

If the polynomial defining the curve has a degree d, any line cuts the curve in at most d points. Bézout's theorem asserts that this number is exactly d, if the points are searched in the projective plane over an algebraically closed field (for example the complex numbers), and counted with their multiplicity. The method of computation that follows proves again this theorem, in this simple case. To compute the intersection of the curve defined by the polynomial p with the line of equation ax+by+c = 0, one solves the equation of the line for x (or for y if a = 0).

In algebraic geometry, the Kempf vanishing theorem, introduced by , states that the higher cohomology group Hi(G/B,L(λ)) (i > 0) vanishes whenever λ is a dominant weight of B. Here G is a reductive algebraic group over an algebraically closed field, B a Borel subgroup, and L(λ) a line bundle associated to λ. In characteristic 0 this is a special case of the Borel–Weil–Bott theorem, but unlike the Borel–Weil–Bott theorem, the Kempf vanishing theorem still holds in positive characteristic. and found simpler proofs of the Kempf vanishing theorem using the Frobenius morphism.

Laurent series are the complex-valued equivalent to Taylor series, but can be used to study the behavior of functions near singularities through infinite sums of more well understood functions, such as polynomials. A bounded function that is holomorphic in the entire complex plane must be constant; this is Liouville's theorem. It can be used to provide a natural and short proof for the fundamental theorem of algebra which states that the field of complex numbers is algebraically closed. If a function is holomorphic throughout a connected domain then its values are fully determined by its values on any smaller subdomain.

Classically the theta characteristics were divided into these two kinds, odd and even, according to the value of the Arf invariant of a certain quadratic form Q with values mod 2. Thus in case of g = 3 and a plane quartic curve, there were 28 of one type, and the remaining 36 of the other; this is basic in the question of counting bitangents, as it corresponds to the 28 bitangents of a quartic. The geometric construction of Q as an intersection form is with modern tools possible algebraically. In fact the Weil pairing applies, in its abelian variety form.

Many of the theorems and results in transcendental number theory concerning the exponential function have analogues involving the modular function j. Writing q = e2πi for the nome and j() = J(q), Daniel Bertrand conjectured that if q1 and q2 are non-zero algebraic numbers in the complex unit disc that are multiplicatively independent, then J(q1) and J(q2) are algebraically independent over the rational numbers.Bertrand, (1997), conjecture 2 in section 5. Although not obviously related to the four exponentials conjecture, Bertrand's conjecture in fact implies a special case known as the weak four exponentials conjecture.

The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Linear actions of Lie groups are especially important, and are studied in representation theory. In the 1940s-1950s, Ellis Kolchin, Armand Borel, and Claude Chevalley realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, as well as in algebraic geometry.

In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed field such as the complex numbers, these fibers are elliptic curves, perhaps without a chosen origin.) This is equivalent to the generic fiber being a smooth curve of genus one. This follows from proper base change. The surface and the base curve are assumed to be non-singular (complex manifolds or regular schemes, depending on the context).

By Bers's theorem, quasi-Fuchsian groups (of some fixed genus) are parameterized by points in T×T, where T is Teichmüller space of the same genus. Suppose that there is a sequence of quasi-Fuchsian groups corresponding to points (gi, hi) in T×T. Also suppose that the sequences gi, hi converge to points μ,μ′ in the Thurston boundary of Teichmüller space of projective measured laminations. If the points μ,μ′ have the property that any nonzero measured lamination has positive intersection number with at least one of them, then the sequence of quasi-Fuchsian groups has a subsequence that converges algebraically.

Modular representation theory was developed by Richard Brauer from about 1940 onwards to study in greater depth the relationships between the characteristic p representation theory, ordinary character theory and structure of G, especially as the latter relates to the embedding of, and relationships between, its p-subgroups. Such results can be applied in group theory to problems not directly phrased in terms of representations. Brauer introduced the notion now known as the Brauer character. When K is algebraically closed of positive characteristic p, there is a bijection between roots of unity in K and complex roots of unity of order prime to p.

From an algebraic perspective, complex numbers enter quite naturally into the study of algebraic functions. First of all, by the fundamental theorem of algebra, the complex numbers are an algebraically closed field. Hence any polynomial relation p(y, x) = 0 is guaranteed to have at least one solution (and in general a number of solutions not exceeding the degree of p in y) for y at each point x, provided we allow y to assume complex as well as real values. Thus, problems to do with the domain of an algebraic function can safely be minimized.

For several parameters, the covariance matrices and information matrices are elements of the convex cone of nonnegative-definite symmetric matrices in a partially ordered vector space, under the Loewner (Löwner) order. This cone is closed under matrix addition and inversion, as well as under the multiplication of positive real numbers and matrices. An exposition of matrix theory and Loewner order appears in Pukelsheim. The traditional optimality criteria are the information matrix's invariants, in the sense of invariant theory; algebraically, the traditional optimality criteria are functionals of the eigenvalues of the (Fisher) information matrix (see optimal design).

For V embedded in a projective space Pn and defined over some algebraically closed field K, the degree d of V is the number of points of intersection of V, defined over K, with a linear subspace L in general position, when :\dim(V) + \dim(L) = n. Here dim(V) is the dimension of V, and the codimension of L will be equal to that dimension. The degree d is an extrinsic quantity, and not intrinsic as a property of V. For example, the projective line has an (essentially unique) embedding of degree n in Pn.

For any commutative ring , denote the ring of symmetric polynomials in the variables with coefficients in by . This is a polynomial ring in the n elementary symmetric polynomials for . (Note that is not among these polynomials; since , it cannot be member of any set of algebraically independent elements.) This means that every symmetric polynomial has a unique representation : P(X_1,\ldots, X_n)=Q\big(e_1(X_1 , \ldots ,X_n), \ldots, e_n(X_1 , \ldots ,X_n)\big) for some polynomial . Another way of saying the same thing is that the ring homomorphism that sends to for defines an isomorphism between and .

When EPSPs and IPSPs are generated simultaneously in the same cell, the output response will be determined by the relative strengths of the excitatory and inhibitory inputs. Output instructions are thus determined by this algebraic processing of information. Because the discharge threshold across a synapse is a function of the presynaptic volleys that act upon it, and because a given neuron may receive branches from many axons, the passage of impulses in a network of such synapses can be highly varied. The versatility of the synapse arises from its ability to modify information by algebraically summing input signals.

There are other similarities to number theory -- error estimates are improved upon, in much the same way that error estimates of the prime number theorem are improved upon. Also, there is a Selberg zeta function which is formally similar to the usual Riemann zeta function and shares many of its properties. Algebraically, prime geodesics can be lifted to higher surfaces in much the same way that prime ideals in the ring of integers of a number field can be split (factored) in a Galois extension. See Covering map and Splitting of prime ideals in Galois extensions for more details.

Given m and n and r < min(m, n), the determinantal variety Y r is the set of all m × n matrices (over a field k) with rank ≤ r. This is naturally an algebraic variety as the condition that a matrix have rank ≤ r is given by the vanishing of all of its (r + 1) × (r + 1) minors. Considering the generic m × n matrix whose entries are algebraically independent variables x i,j, these minors are polynomials of degree r + 1\. The ideal of k[x i,j] generated by these polynomials is a determinantal ideal.

The F-isocrystal Es/r has a basis over K of the form v, Fv, F2v,...,Fr−1v for some element v, and Frv = psv. The rational number s/r is called the slope of the F-isocrystal. Over a non-algebraically closed field k the simple F-isocrystals are harder to describe explicitly, but an F-isocrystal can still be written as a direct sum of subcrystals that are isoclinic, where an F-crystal is called isoclinic if over the algebraic closure of k it is a sum of F-isocrystals of the same slope.

For applications to number theory, van der Waerden and Weil formulated algebraic geometry over any field, not necessarily algebraically closed. Weil was the first to define an abstract variety (not embedded in projective space), by gluing affine varieties along open subsets, on the model of manifolds in topology. He needed this generality for his construction of the Jacobian variety of a curve over any field. (Later, Jacobians were shown to be projective varieties by Weil, Chow and Matsusaka.) The algebraic geometers of the Italian school had often used the somewhat foggy concept of the generic point of an algebraic variety.

While is its own additive inverse the additive inverse of a positive number is negative, and the additive inverse of a negative number is positive. A double application of this operation is written as The plus sign is predominantly used in algebra to denote the binary operation of addition, and only rarely to emphasize the positivity of an expression. In common numeral notation (used in arithmetic and elsewhere), the sign of a number is often made explicit by placing a plus or a minus sign before the number. For example, denotes "positive three", and denotes "negative three" (algebraically: the additive inverse of ).

Parabola (, red) and cubic (, blue) in projective space Just as the formulas for the roots of second, third, and fourth degree polynomials suggest extending real numbers to the more algebraically complete setting of the complex numbers, many properties of algebraic varieties suggest extending affine space to a more geometrically complete projective space. Whereas the complex numbers are obtained by adding the number i, a root of the polynomial , projective space is obtained by adding in appropriate points "at infinity", points where parallel lines may meet. To see how this might come about, consider the variety . If we draw it, we get a parabola.

The dimension of a manifold depends on the base field with respect to which Euclidean space is defined. While analysis usually assumes a manifold to be over the real numbers, it is sometimes useful in the study of complex manifolds and algebraic varieties to work over the complex numbers instead. A complex number (x + iy) has a real part x and an imaginary part y, where x and y are both real numbers; hence, the complex dimension is half the real dimension. Conversely, in algebraically unconstrained contexts, a single complex coordinate system may be applied to an object having two real dimensions.

Just as in the case of nimber addition, there is a means of computing the nimber product of finite ordinals. This is determined by the rules that # The nimber product of a Fermat 2-power (numbers of the form ) with a smaller number is equal to their ordinary product; # The nimber square of a Fermat 2-power is equal to as evaluated under the ordinary multiplication of natural numbers. The smallest algebraically closed field of nimbers is the set of nimbers less than the ordinal , where is the smallest infinite ordinal. It follows that as a nimber, is transcendental over the field.

Changing the definition of "well known" to include additional functions can change the set of equations with closed-form solutions. Many cumulative distribution functions cannot be expressed in closed form, unless one considers special functions such as the error function or gamma function to be well known. It is possible to solve the quintic equation if general hypergeometric functions are included, although the solution is far too complicated algebraically to be useful. For many practical computer applications, it is entirely reasonable to assume that the gamma function and other special functions are well known since numerical implementations are widely available.

In mathematics, the irrelevant ideal is the ideal of a graded ring generated by the homogeneous elements of degree greater than zero. More generally, a homogeneous ideal of a graded ring is called an irrelevant ideal if its radical contains the irrelevant ideal. The terminology arises from the connection with algebraic geometry. If R = k[x0, ..., xn] (a multivariate polynomial ring in n+1 variables over an algebraically closed field k) graded with respect to degree, there is a bijective correspondence between projective algebraic sets in projective n-space over k and homogeneous, radical ideals of R not equal to the irrelevant ideal.

In the 1730s, he first established and used what was later to be known as Catalan numbers.The 18th century Chinese discovery of the Catalan numbers The Jesuit missionaries' influence can be seen by many traces of European mathematics in his works, including the use of Euclidean notions of continuous proportions, series addition, subtraction, multiplication and division, series reversion, and the binomial theorem. Minggatu's work is remarkable in that expansions in series, trigonometric and logarithmic were apprehended algebraically and inductively without the aid of differential and integral calculus. In 1742 he participated in the revision of the Compendium of Observational and Computational Astronomy.

A cubic curve may have a singular point, in which case it has a parametrization in terms of a projective line. Otherwise a non-singular cubic curve is known to have nine points of inflection, over an algebraically closed field such as the complex numbers. This can be shown by taking the homogeneous version of the Hessian matrix, which defines again a cubic, and intersecting it with C; the intersections are then counted by Bézout's theorem. However, only three of these points may be real, so that the others cannot be seen in the real projective plane by drawing the curve.

2 and 10.1.1. In particular, simple groups over an algebraically closed field k are classified (up to quotients by finite central subgroup schemes) by their Dynkin diagrams. It is striking that this classification is independent of the characteristic of k. For example, the exceptional Lie groups G2, F4, E6, E7, and E8 can be defined in any characteristic (and even as group schemes over Z). The classification of finite simple groups says that most finite simple groups arise as the group of k-points of a simple algebraic group over a finite field k, or as minor variants of that construction.

For algebraically closed fields of characteristic p>0 Lie's theorem holds provided the dimension of the representation is less than p (see the proof below), but can fail for representations of dimension p. An example is given by the 3-dimensional nilpotent Lie algebra spanned by 1, x, and d/dx acting on the p-dimensional vector space k[x]/(xp), which has no eigenvectors. Taking the semidirect product of this 3-dimensional Lie algebra by the p-dimensional representation (considered as an abelian Lie algebra) gives a solvable Lie algebra whose derived algebra is not nilpotent.

In general, there is no difference either algebraically or logically between the homogeneous coordinates of points and lines. So plane geometry with points as the fundamental elements and plane geometry with lines as the fundamental elements are equivalent except for interpretation. This leads to the concept of duality in projective geometry, the principle that the roles of points and lines can be interchanged in a theorem in projective geometry and the result will also be a theorem. Analogously, the theory of points in projective 3-space is dual to the theory of planes in projective 3-space, and so on for higher dimensions.

The International Data Encryption Algorithm (IDEA) is a block cipher designed by James Massey of ETH Zurich and Xuejia Lai; it was first described in 1991, as an intended replacement for DES. IDEA operates on 64-bit blocks using a 128-bit key, and consists of a series of eight identical transformations (a round) and an output transformation (the half-round). The processes for encryption and decryption are similar. IDEA derives much of its security by interleaving operations from different groups – modular addition and multiplication, and bitwise exclusive or (XOR) – which are algebraically "incompatible" in some sense.

In algebraic geometry, a morphism of schemes :f: X -> Y is called radicial or universally injective, if, for every field K the induced map X(K) → Y(K) is injective. (EGA I, (3.5.4)) This is a generalization of the notion of a purely inseparable extension of fields (sometimes called a radicial extension, which should not be confused with a radical extension.) It suffices to check this for K algebraically closed. This is equivalent to the following condition: f is injective on the topological spaces and for every point x in X, the extension of the residue fields :k(f(x)) ⊂ k(x) is radicial, i.e.

Macaulay's resultant, named after Francis Sowerby Macaulay, also called the multivariate resultant, or the multipolynomial resultant,, Chapter 3. Resultants is a generalization of the homogeneous resultant to homogeneous polynomials in indeterminates. Macaulay's resultant is a polynomial in the coefficients of these homogeneous polynomials that vanishes if and only if the polynomials have a common non-zero solution in an algebraically closed field containing the coefficients, or, equivalently, if the hyper surfaces defined by the polynomials have a common zero in the dimensional projective space. The multivariate resultant is, with Gröbner bases, one of the main tools of effective elimination theory (elimination theory on computers).

The determinant of this matrix is the U-resultant. As with the original U-resultant, it is zero if and only if P_1, \ldots, P_k have infinitely many common projective zeros (that is if the projective algebraic set defined by P_1, \ldots, P_k has infinitely many points over an algebraic closure of ). Again as with the original U-resultant, when this U-resultant is not zero, it factorizes into linear factors over any algebraically closed extension of . The coefficients of these linear factors are the homogeneous coordinates of the common zeros of P_1, \ldots, P_k, and the multiplicity of a common zero equals the multiplicity of the corresponding linear factor.

A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations where the are polynomials in several variables, say , over some field . A solution of a polynomial system is a set of values for the s which belong to some algebraically closed field extension of , and make all equations true. When is the field of rational numbers, is generally assumed to be the field of complex numbers, because each solution belongs to a field extension of , which is isomorphic to a subfield of the complex numbers. This article is about the methods for solving, that is, finding all solutions or describing them.

This is algebraic over and is the correct global Frobenius image in terms of the embedding of into ; moreover, the coefficients are algebraic and the result can be expressed algebraically. However, they are of degree 120, the order of the Galois group, illustrating the fact that explicit computations are much more easily accomplished if -adic results will suffice. If is an abelian extension of global fields, we get a much stronger congruence since it depends only on the prime in the base field . For an example, consider the extension of obtained by adjoining a root satisfying :\beta^5+\beta^4-4\beta^3-3\beta^2+3\beta+1=0 to .

In economics and consumer theory, quasilinear utility functions are linear in one argument, generally the numeraire. Quasilinear preferences can be represented by the utility function u(x_1, x_2, \ldots, x_n) = x_1 + \theta (x_2, \ldots, x_n) where \theta is strictly concave. A useful property of the quasilinear utility function is that the Marshallian/Walrasian demand for x_2, \ldots, x_n does not depend on wealth and is thus not subject to a wealth effect; The absence of a wealth effect simplifies analysis and makes quasilinear utility functions a common choice for modelling. Furthermore, when utility is quasilinear, compensating variation (CV), equivalent variation (EV), and consumer surplus are algebraically equivalent.

Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or a number field, the classification is well understood. The classification of finite simple groups says that most finite simple groups arise as the group G(k) of k-rational points of a simple algebraic group G over a finite field k, or as minor variants of that construction.

The other term, , gives the distance the zeros are away from the axis of symmetry, where the plus sign represents the distance to the right, and the minus sign represents the distance to the left. If this distance term were to decrease to zero, the value of the axis of symmetry would be the value of the only zero, that is, there is only one possible solution to the quadratic equation. Algebraically, this means that , or simply (where the left-hand side is referred to as the discriminant). This is one of three cases, where the discriminant indicates how many zeros the parabola will have.

If K is a perfect field --- for example a field of characteristic zero, or a finite field, or an algebraically closed field --- then every extension of K is separable so that separable K-algebras are finite products of matrix algebras over finite-dimensional division algebras over field K. In other words, if K is a perfect field, there is no difference between a separable algebra over K and a finite-dimensional semisimple algebra over K. It can be shown by a generalized theorem of Maschke that an associative K-algebra A is separable if for every field extension \scriptstyle L/K the algebra \scriptstyle A\otimes_K L is semisimple.

In the case of multivariate normal distributions, the parameters would be n − 1 correlations and (n − 1)(n − 2)/2 partial correlations, which were noted to be algebraically independent in (−1, 1). An entirely different motivation underlay the first formal definition of vines in Cooke. Uncertainty analyses of large risk models, such as those undertaken for the European Union and the US Nuclear Regulatory Commission for accidents at nuclear power plants, involve quantifying and propagating uncertainty over hundreds of variables. Dependence information for such studies had been captured with Markov trees, which are trees constructed with nodes as univariate random variables and edges as bivariate copulas.

The transcendence of and are direct corollaries of this theorem. Suppose is a non-zero algebraic number; then is a linearly independent set over the rationals, and therefore by the first formulation of the theorem is an algebraically independent set; or in other words is transcendental. In particular, is transcendental. (A more elementary proof that is transcendental is outlined in the article on transcendental numbers.) Alternatively, by the second formulation of the theorem, if is a non-zero algebraic number, then is a set of distinct algebraic numbers, and so the set is linearly independent over the algebraic numbers and in particular cannot be algebraic and so it is transcendental.

Nilpotent orbits form a partially ordered set: given two nilpotent orbits, O1 is less than or equal to O2 if O1 is contained in the Zariski closure of O2. This poset has a unique minimal element, zero orbit, and unique maximal element, the regular nilpotent orbit, but in general, it is not a graded poset. If the ground field is algebraically closed then the zero orbit is covered by a unique orbit, called the minimal orbit, and the regular orbit covers a unique orbit, called the subregular orbit. In the case of the special linear group SLn, the nilpotent orbits are parametrized by the partitions of n.

In the special case when Y is the spectrum of an algebraically closed field (a point), Rqf∗(F ) is the same as Hq(F ). Suppose that X is a Noetherian scheme. An abelian étale sheaf F over X is called finite locally constant if it is represented by an étale cover of X. It is called constructible if X can be covered by a finite family of subschemes on each of which the restriction of F is finite locally constant. It is called torsion if F(U) is a torsion group for all étale covers U of X. Finite locally constant sheaves are constructible, and constructible sheaves are torsion.

For an algebraically closed field k, a matrix g in GL(n,k) is called semisimple if it is diagonalizable, and unipotent if the matrix g − 1 is nilpotent. Equivalently, g is unipotent if all eigenvalues of g are equal to 1. The Jordan canonical form for matrices implies that every element g of GL(n,k) can be written uniquely as a product g = gssgu such that gss is semisimple, gu is unipotent, and gss and gu commute with each other. For any field k, an element g of GL(n,k) is said to be semisimple if it becomes diagonalizable over the algebraic closure of k.

In classical algebraic geometry (that is, the part of algebraic geometry in which one does not use schemes, which were introduced by Grothendieck around 1960), the Zariski topology is defined on algebraic varieties. The Zariski topology, defined on the points of the variety, is the topology such that the closed sets are the algebraic subsets of the variety. As the most elementary algebraic varieties are affine and projective varieties, it is useful to make this definition more explicit in both cases. We assume that we are working over a fixed, algebraically closed field k (in classical geometry k is almost always the complex numbers).

Strong minimality was one of the early notions in the new field of classification theory and stability theory that was opened up by Morley's theorem on totally categorical structures. The nontrivial standard examples for strongly minimal theories are the one-sorted theories of infinite-dimensional vector spaces, and the theories ACFp of algebraically closed fields. As the example ACFp shows, the parametrically definable subsets of the square of the domain of a minimal structure can be relatively complicated ("curves"). More generally, a subset of a structure that is defined as the set of realizations of a formula φ(x) is called a minimal set if every parametrically definable subset of it is either finite or cofinite.

Kepler's equation is where is the mean anomaly, is the eccentric anomaly, and is the eccentricity. The 'eccentric anomaly' is useful to compute the position of a point moving in a Keplerian orbit. As for instance, if the body passes the periastron at coordinates , , at time , then to find out the position of the body at any time, you first calculate the mean anomaly from the time and the mean motion by the formula , then solve the Kepler equation above to get , then get the coordinates from: where is the semi-major axis, the semi-minor axis. Kepler's equation is a transcendental equation because sine is a transcendental function, meaning it cannot be solved for algebraically.

Given any complex numbers (called coefficients) , the equation :a_n z^n + \dotsb + a_1 z + a_0 = 0 has at least one complex solution z, provided that at least one of the higher coefficients is nonzero. This is the statement of the fundamental theorem of algebra, of Carl Friedrich Gauss and Jean le Rond d'Alembert. Because of this fact, C is called an algebraically closed field. This property does not hold for the field of rational numbers Q (the polynomial does not have a rational root, since is not a rational number) nor the real numbers R (the polynomial does not have a real root for , since the square of is positive for any real number ).

In fact, most of the important Lie groups (but not all) can be expressed as matrix groups. If this idea is generalised to matrices with complex numbers as entries, then we get further useful Lie groups, such as the unitary group U(n). We can also consider matrices with quaternions as entries; in this case, there is no well-defined notion of a determinant (and thus no good way to define a quaternionic "volume"), but we can still define a group analogous to the orthogonal group, the symplectic group Sp(n). Furthermore, the idea can be treated purely algebraically with matrices over any field, but then the groups are not Lie groups.

The definition of the dual depends on the choice of embedding of the graph , so it is a property of plane graphs (graphs that are already embedded in the plane) rather than planar graphs (graphs that may be embedded but for which the embedding is not yet known). For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph. Historically, the first form of graph duality to be recognized was the association of the Platonic solids into pairs of dual polyhedra. Graph duality is a topological generalization of the geometric concepts of dual polyhedra and dual tessellations, and is in turn generalized algebraically by the concept of a dual matroid.

The Boolean domain {0, 1} can be replaced by the unit interval , in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1-x, conjunction (AND) is replaced with multiplication (xy), and disjunction (OR) is defined via De Morgan's law to be 1-(1-x)(1-y). Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.

A group scheme G over a noetherian scheme S is finite and flat if and only if OG is a locally free OS-module of finite rank. The rank is a locally constant function on S, and is called the order of G. The order of a constant group scheme is equal to the order of the corresponding group, and in general, order behaves well with respect to base change and finite flat restriction of scalars. Among the finite flat group schemes, the constants (cf. example above) form a special class, and over an algebraically closed field of characteristic zero, the category of finite groups is equivalent to the category of constant finite group schemes.

The group G(m, p, n) is an index-p subgroup of G(m, 1, n). G(m, p, n) is of order mnn!/p. As matrices, it may be realized as the subset in which the product of the nonzero entries is an (m/p)th root of unity (rather than just an mth root). Algebraically, G(m, p, n) is a semidirect product of an abelian group of order mn/p by the symmetric group Sym(n); the elements of the abelian group are of the form (θa1, θa2, ..., θan), where θ is a primitive mth root of unity and ∑ai ≡ 0 mod p, and Sym(n) acts by permutations of the coordinates.

This ratio is the numerical value of the quantity or the number of units contained in the quantity. The definition of the metre per second above satisfies this requirement since it, together with the definition of velocity, implies that v/mps = (d/m)/(t/s); thus if the ratios of distance and time to their units are determined, then so is the ratio of velocity to its unit. The definition, by itself, is inadequate since it only determines the ratio in one specific case; it may be thought of as exhibiting a specimen of the unit. A new coherent unit cannot be defined merely by expressing it algebraically in terms of already defined units.

A quaternion algebra over a field F is a four-dimensional central simple F-algebra. A quaternion algebra has a basis 1, i, j, ij where i^2, j^2 \in F^\times and ij = -ji. A quaternion algebra is said to be split over F if it is isomorphic as an F-algebra to the algebra of matrices M_2(F); a quaternion algebra over an algebraically closed field is always split. If \sigma is an embedding of F into a field E we shall denote by A \otimes_\sigma E the algebra obtained by extending scalars from F to E where we view F as a subfield of E via \sigma.

In logic, the unit interval [0,1] can be interpreted as a generalization of the Boolean domain {0,1}, in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1-x ; conjunction (AND) is replaced with multiplication (xy); and disjunction (OR) is defined, per De Morgan's laws, as 1-(1-x)(1-y) . Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.

What is true for the generic point is true for "most" points of the variety. In Weil's Foundations of Algebraic Geometry (1946), generic points are constructed by taking points in a very large algebraically closed field, called a universal domain. Although this worked as a foundation, it was awkward: there were many different generic points for the same variety. (In the later theory of schemes, each algebraic variety has a single generic point.) In the 1950s, Claude Chevalley, Masayoshi Nagata and Jean-Pierre Serre, motivated in part by the Weil conjectures relating number theory and algebraic geometry, further extended the objects of algebraic geometry, for example by generalizing the base rings allowed.

In selling the product, Metcalfe argued that customers needed Ethernet cards to grow above a certain critical mass if they were to reap the benefits of their network. According to Metcalfe, the rationale behind the sale of networking cards was that the cost of the network was directly proportional to the number of cards installed, but the value of the network was proportional to the square of the number of users. This was expressed algebraically as having a cost of N, and a value of N2. While the actual numbers behind this proposition were never firm, the concept allowed customers to share access to expensive resources like disk drives and printers, send e-mail, and eventually access the Internet.

Geometrically, this means that the graph has no cusps, self-intersections, or isolated points. Algebraically, this holds if and only if the discriminant : \Delta = -16(4a^3 + 27b^2) is not equal to zero. (Although the factor −16 is irrelevant to whether or not the curve is non- singular, this definition of the discriminant is useful in a more advanced study of elliptic curves.) The (real) graph of a non-singular curve has two components if its discriminant is positive, and one component if it is negative. For example, in the graphs shown in figure to the right, the discriminant in the first case is 64, and in the second case is −368.

In mathematics, the gonality of an algebraic curve C is defined as the lowest degree of a nonconstant rational map from C to the projective line. In more algebraic terms, if C is defined over the field K and K(C) denotes the function field of C, then the gonality is the minimum value taken by the degrees of field extensions :K(C)/K(f) of the function field over its subfields generated by single functions f. If K is algebraically closed, then the gonality is 1 precisely for curves of genus 0. The gonality is 2 for curves of genus 1 (elliptic curves) and for hyperelliptic curves (this includes all curves of genus 2).

The resulting configuration, the Hesse configuration, shares with the Möbius–Kantor configuration the property of being realizable with complex coordinates but not with real coordinates.. Deleting any one point from the Hesse configuration produces a copy of the Möbius–Kantor configuration. Both configurations may also be described algebraically in terms of the abelian group \Z_3\times \Z_3 with nine elements. This group has four subgroups of order three (the subsets of elements of the form (i,0), (i,i), (i,2i), and (0,i) respectively), each of which can be used to partition the nine group elements into three cosets of three elements per coset. These nine elements and twelve cosets form the Hesse configuration.

Demonstration, with Cuisenaire rods, that 1, 2, 8, 9, and 12 are refactorable A refactorable number or tau number is an integer n that is divisible by the count of its divisors, or to put it algebraically, n is such that \tau(n)\mid n. The first few refactorable numbers are listed in as :1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, ... For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6. There are infinitely many refactorable numbers.

The first example of a K3 surface with Picard number 22 was given by , who observed that the Fermat quartic :w4 \+ x4 \+ y4 \+ z4 = 0 has Picard number 22 over algebraically closed fields of characteristic 3 mod 4. Then Shioda showed that the elliptic modular surface of level 4 (the universal generalized elliptic curve E(4) → X(4)) in characteristic 3 mod 4 is a K3 surface with Picard number 22, as is the Kummer surface of the product of two supersingular elliptic curves in odd characteristic. showed that all K3 surfaces with Picard number 22 are double covers of the projective plane. In the case of characteristic 2 the double cover may need to be an inseparable covering.

A (non- degenerate) conic is completely determined by five points in general position (no three collinear) in a plane and the system of conics which pass through a fixed set of four points (again in a plane and no three collinear) is called a pencil of conics.. The four common points are called the base points of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles. In a projective plane defined over an algebraically closed field any two conics meet in four points (counted with multiplicity) and so, determine the pencil of conics based on these four points.

In three papers written between 1969 and 1976 (the last two in collaboration with Enrico Bombieri), Mumford extended the Enriques–Kodaira classification of smooth projective surfaces from the case of the complex ground field to the case of an algebraically closed ground field of characteristic p. The final answer turns out to be essentially the same as the answer in the complex case (though the methods employed are sometimes quite different), once two important adjustments are made. The first is that one may get "non-classical" surfaces, which come about when p-torsion in the Picard scheme degenerates to a non- reduced group scheme. The second is the possibility of obtaining quasi- elliptic surfaces in characteristics two and three.

The latter term (degree absolute), which was the unit's official name from 1948 until 1954, was ambiguous since it could also be interpreted as referring to the Rankine scale. Before the 13th CGPM, the plural form was "degrees absolute". The 13th CGPM changed the unit name to simply "kelvin" (symbol: K). The omission of "degree" indicates that it is not relative to an arbitrary reference point like the Celsius and Fahrenheit scales (although the Rankine scale continued to use "degree Rankine"), but rather an absolute unit of measure which can be manipulated algebraically (e.g. multiplied by two to indicate twice the amount of "mean energy" available among elementary degrees of freedom of the system).

For any central simple algebra A over a field K, the period of A divides the index of A, and the two numbers have the same prime factors.Gille & Szamuely (2006), Proposition 4.5.13. The period-index problem is to bound the index in terms of the period, for fields K of interest. For example, if A is a central simple algebra over a local field or global field, then Albert–Brauer–Hasse–Noether showed that the index of A is equal to the period of A. For a central simple algebra A over a field K of transcendence degree n over an algebraically closed field, it is conjectured that ind(A) divides per(A)n−1.

Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be solved algebraically). This can be done for all such equations of degree one, two, three, or four; but for degree five or more it can only be done for some equations, not for all. A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation (see Root-finding algorithm) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations).

The fundamental theorem of algebra states that the field of the complex numbers is closed algebraically, that is, all polynomial equations with complex coefficients and degree at least one have a solution. It follows that all polynomial equations of degree 1 or more with real coefficients have a complex solution. On the other hand, an equation such as x^2 + 1 = 0 does not have a solution in \R (the solutions are the imaginary units and ). While the real solutions of real equations are intuitive (they are the -coordinates of the points where the curve intersects the -axis), the existence of complex solutions to real equations can be surprising and less easy to visualize.

Bézout's theorem predicts that the number of points of intersection of two curves is equal to the product of their degrees (assuming an algebraically closed field and with certain conventions followed for counting intersection multiplicities). Bézout's theorem predicts there is one point of intersection of two lines and in general this is true, but when the lines are parallel the point of intersection is infinite. Homogeneous coordinates are used to locate the point of intersection in this case. Similarly, Bézout's theorem predicts that a line will intersect a conic at two points, but in some cases one or both of the points is infinite and homogeneous coordinates must be used to locate them.

A k-morphism is a regular function between k-algebraic sets whose defining polynomials' coefficients belong to k. One reason for considering the zero-locus in An(kalg) and not An(k) is that, for two distinct k-algebraic sets X1 and X2, the intersections X1∩An(k) and X2∩An(k) can be identical; in fact, the zero-locus in An(k) of any subset of k[x1, …, xn] is the zero-locus of a single element of k[x1, …, xn] if k is not algebraically closed. A k-variety is called a variety if it is absolutely irreducible, i.e. is not the union of two strictly smaller kalg-algebraic sets.

In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a meet (or greatest lower bound) for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the inverse order and vice versa. Semilattices can also be defined algebraically: join and meet are associative, commutative, idempotent binary operations, and any such operation induces a partial order (and the respective inverse order) such that the result of the operation for any two elements is the least upper bound (or greatest lower bound) of the elements with respect to this partial order.

In the foundational approach of André Weil, developed in his Foundations of Algebraic Geometry, generic points played an important role, but were handled in a different manner. For an algebraic variety V over a field K, generic points of V were a whole class of points of V taking values in a universal domain Ω, an algebraically closed field containing K but also an infinite supply of fresh indeterminates. This approach worked, without any need to deal directly with the topology of V (K-Zariski topology, that is), because the specializations could all be discussed at the field level (as in the valuation theory approach to algebraic geometry, popular in the 1930s). This was at a cost of there being a huge collection of equally generic points.

An isomorphism between a group G and the fundamental group of a graph of groups is called a splitting of G. If the edge groups in the splitting come from a particular class of groups (e.g. finite, cyclic, abelian, etc.), the splitting is said to be a splitting over that class. Thus a splitting where all edge groups are finite is called a splitting over finite groups. Algebraically, a splitting of G with trivial edge groups corresponds to a free product decomposition :G=(\ast A_v)\ast F(X) where F(X) is a free group with free basis X = E+(A−T) consisting of all positively oriented edges (with respect to some orientation on A) in the complement of some spanning tree T of A.

Algebra being a fundamental tool in any area amenable to mathematical treatment, these considerations combine to make the algebra of two values of fundamental importance to computer hardware, mathematical logic, and set theory. Two-valued logic can be extended to multi- valued logic, notably by replacing the Boolean domain {0, 1} with the unit interval [0,1], in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1 − x, conjunction (AND) is replaced with multiplication (xy), and disjunction (OR) is defined via De Morgan's law. Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic.

The isopleth crosses the scale for T at just under 4.65; a larger figure printed in high resolution on paper would yield T = 4.64 to three-digit precision. Note that any variable can be calculated from values of the other two, a feature of nomograms that is particularly useful for equations in which a variable cannot be algebraically isolated from the other variables. Straight scales are useful for relatively simple calculations, but for more complex calculations the use of simple or elaborate curved scales may be required. Nomograms for more than three variables can be constructed by incorporating a grid of scales for two of the variables, or by concatenating individual nomograms of fewer numbers of variables into a compound nomogram.

Bézout's identity works for univariate polynomials over a field exactly in the same ways as for integers. In particular the Bézout's coefficients and the greatest common divisor may be computed with the extended Euclidean algorithm. As the common roots of two polynomials are the roots of their greatest common divisor, Bézout's identity and fundamental theorem of algebra imply the following result: :For univariate polynomials f and g with coefficients in a field, there exist polynomials a and b such that af + bg = 1 if and only if f and g have no common root in any algebraically closed field (commonly the field of complex numbers). The generalization of this result to any number of polynomials and indeterminates is Hilbert's Nullstellensatz.

However, the symmetric difference of two Eulerian subgraphs (the graph consisting of the edges that belong to exactly one of the two given graphs) is again Eulerian. This follows from the fact that the symmetric difference of two sets with an even number of elements is also even. Applying this fact separately to the neighborhoods of each vertex shows that the symmetric difference operator preserves the property of being Eulerian. A family of sets closed under the symmetric difference operation can be described algebraically as a vector space over the two-element finite field \Z_2.. This field has two elements, 0 and 1, and its addition and multiplication operations can be described as the familiar addition and multiplication of integers, taken modulo 2\.

In another direction, that of torsors, these were already implicit in the infinite descent arguments of Fermat for elliptic curves. Numerous direct calculations were done, and the proof of the Mordell–Weil theorem had to proceed by some surrogate of a finiteness proof for a particular H1 group. The 'twisted' nature of objects over fields that are not algebraically closed, which are not isomorphic but become so over the algebraic closure, was also known in many cases linked to other algebraic groups (such as quadratic forms, simple algebras, Severi–Brauer varieties), in the 1930s, before the general theory arrived. The needs of number theory were in particular expressed by the requirement to have control of a local-global principle for Galois cohomology.

An algebraic plane curve is a curve in an affine or projective plane given by one polynomial equation (or , where F is a homogeneous polynomial, in the projective case.) Algebraic curves have been studied extensively since the 18th century. Every algebraic plane curve has a degree, the degree of the defining equation, which is equal, in case of an algebraically closed field, to the number of intersections of the curve with a line in general position. For example, the circle given by the equation has degree 2. The non-singular plane algebraic curves of degree 2 are called conic sections, and their projective completion are all isomorphic to the projective completion of the circle (that is the projective curve of equation ).

Wherever eigenvalues are considered, as these are roots of a polynomial they may exist only in a larger field than that of the entries of the matrix; for instance they may be complex in case of a matrix with real entries. The possibility to reinterpret the entries of a matrix as elements of a larger field (for example, to view a real matrix as a complex matrix whose entries happen to be all real) then allows considering each square matrix to possess a full set of eigenvalues. Alternatively one can consider only matrices with entries in an algebraically closed field, such as C, from the outset. More generally, matrices with entries in a ring R are widely used in mathematics.

To put the above informal discussion of braid groups on firm ground, one needs to use the homotopy concept of algebraic topology, defining braid groups as fundamental groups of a configuration space. Alternatively, one can define the braid group purely algebraically via the braid relations, keeping the pictures in mind only to guide the intuition. To explain how to reduce a braid group in the sense of Artin to a fundamental group, we consider a connected manifold X of dimension at least 2. The symmetric product of n copies of X means the quotient of X^n, the n-fold Cartesian product of X by the permutation action of the symmetric group on n strands operating on the indices of coordinates.

This is the explicit form in this case of the abstract result that over an algebraically closed field K (such as the complex numbers) the regular representation of G is completely reducible, provided that the characteristic of K (if it is a prime number p) doesn't divide the order of G. That is called Maschke's theorem. In this case the condition on the characteristic is implied by the existence of a primitive n-th root of unity, which cannot happen in the case of prime characteristic p dividing n. Circulant determinants were first encountered in nineteenth century mathematics, and the consequence of their diagonalisation drawn. Namely, the determinant of a circulant is the product of the n eigenvalues for the n eigenvectors described above.

As a reaction to claims by the British that they were governing with "the true interests of the natives at heart", he wrote: "The dimensions of "the true interests of the natives at heart" are algebraically equal to the length, breadth and depth of the whiteman's pocket." In 1908 he exposed European corruption in the handling of railway finances and in 1919 he argued successfully for the chiefs whose land had been taken by the British in front of the Privy Council in London. As a result, the colonial government was forced to pay compensation to the chiefs. In 1909, he came out publicly against the prohibition of spirits into Nigeria which he felt will ultimately lead to reduced government revenues and thereafter increased taxation.

Binary relations over sets X and Y can be represented algebraically by logical matrices indexed by X and Y with entries in the Boolean semiring (addition corresponds to OR and multiplication to AND) where matrix addition corresponds to union of relations, matrix multiplication corresponds to composition of relations (of a relation over X and Y and a relation over Y and Z), the Hadamard product corresponds to intersection of relations, the zero matrix corresponds to the empty relation, and the matrix of ones corresponds to the universal relation. Homogeneous relations (when ) form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring) where the identity matrix corresponds to the identity relation.Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series.

The converse is not true however: there are infinite extensions which are algebraic. For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers. If a is algebraic over K, then K[a], the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. The converse is true as well, if K[a] is a field, then a is algebraic over K. In the special case where K = Q is the field of rational numbers, Q[a] is an example of an algebraic number field. A field with no proper algebraic extensions is called algebraically closed.

In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative rings are like vector bundles on compact spaces". The two precise formulations of the theorems differ somewhat. The original theorem, as stated by Jean-Pierre Serre in 1955, is more algebraic in nature, and concerns vector bundles on an algebraic variety over an algebraically closed field (of any characteristic). The complementary variant stated by Richard Swan in 1962 is more analytic, and concerns (real, complex, or quaternionic) vector bundles on a smooth manifold or Hausdorff space.

The nimber multiplicative inverse of the nonzero ordinal is given by , where is the smallest set of ordinals (nimbers) such that # 0 is an element of ; # if and is an element of , then is also an element of . For all natural numbers , the set of nimbers less than form the Galois field of order . In particular, this implies that the set of finite nimbers is isomorphic to the direct limit as of the fields . This subfield is not algebraically closed, since no other field (so with not a power of 2) is contained in any of those fields, and therefore not in their direct limit; for instance the polynomial , which has a root in , does not have a root in the set of finite nimbers.

In fact, a matrix A over a field containing all of the eigenvalues of A (for example, any matrix over an algebraically closed field) is similar to a triangular matrix. This can be proven by using induction on the fact that A has an eigenvector, by taking the quotient space by the eigenvector and inducting to show that A stabilises a flag, and is thus triangularizable with respect to a basis for that flag. A more precise statement is given by the Jordan normal form theorem, which states that in this situation, A is similar to an upper triangular matrix of a very particular form. The simpler triangularization result is often sufficient however, and in any case used in proving the Jordan normal form theorem.

Venn diagram of information theoretic measures for three variables x, y, and z, represented by the lower left, lower right, and upper circles, respectively. The multivariate mutual information is represented by gray region. Since it may be negative, the areas on the diagram represent signed measures. In information theory there have been various attempts over the years to extend the definition of mutual information to more than two random variables. The expression and study of multivariate higher-degree mutual- information was achieved in two seemingly independent works: McGill (1954) who called these functions “interaction information”, and Hu Kuo Ting (1962) who also first proved the possible negativity of mutual-information for degrees higher than 2 and justified algebraically the intuitive correspondence to Venn diagrams .

In classical algebraic geometry, all varieties were by definition quasi-projective varieties, meaning that they were open subvarieties of closed subvarieties of projective space. For example, in Chapter 1 of Hartshorne a variety over an algebraically closed field is defined to be a quasi-projective variety, but from Chapter 2 onwards, the term variety (also called an abstract variety) refers to a more general object, which locally is a quasi-projective variety, but when viewed as a whole is not necessarily quasi-projective; i.e. it might not have an embedding into projective space. So classically the definition of an algebraic variety required an embedding into projective space, and this embedding was used to define the topology on the variety and the regular functions on the variety.

Methods of computing square roots are numerical analysis algorithms for finding the principal, or non-negative, square root (usually denoted , , or S1/2) of a real number. Arithmetically, it means given S, a procedure for finding a number which when multiplied by itself, yields S; algebraically, it means a procedure for finding the non-negative root of the equation x2 \- S = 0; geometrically, it means given the area of a square, a procedure for constructing a side of the square. Every real number has two square roots. In addition to the principal square root, there is a negative square root equal in magnitude but opposite in sign to the principal square root, except for zero, which has double square roots of zero.

In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it usually means a sequence defined by the same recurrence relation as the Fibonacci numbers, but with different initial conditions causing all members of the sequence to be composite numbers that do not all have a common divisor. To put it algebraically, a sequence of this type is defined by an appropriate choice of two composite numbers a1 and a2, such that the greatest common divisor GCD(a1,a2) is equal to 1, and such that for n > 2 there are no primes in the sequence of numbers calculated from the formula :an = an − 1 + an − 2. The first primefree sequence of this type was published by Ronald Graham in 1964.

An extension L which is a splitting field for a set of polynomials p(X) over K is called a normal extension of K. Given an algebraically closed field A containing K, there is a unique splitting field L of p between K and A, generated by the roots of p. If K is a subfield of the complex numbers, the existence is immediate. On the other hand, the existence of algebraic closures in general is often proved by 'passing to the limit' from the splitting field result, which therefore requires an independent proof to avoid circular reasoning. Given a separable extension K′ of K, a Galois closure L of K′ is a type of splitting field, and also a Galois extension of K containing K′ that is minimal, in an obvious sense.

In the limit the time interval tends to zero, the accelerated frame will rotate at every instant, so the accelerated frame rotates with an angular velocity. The precession can be understood geometrically as a consequence of the fact that the space of velocities in relativity is hyperbolic, and so parallel transport of a vector (the gyroscope's angular velocity) around a circle (its linear velocity) leaves it pointing in a different direction, or understood algebraically as being a result of the non-commutativity of Lorentz transformations. Thomas precession gives a correction to the spin–orbit interaction in quantum mechanics, which takes into account the relativistic time dilation between the electron and the nucleus of an atom. Thomas precession is a kinematic effect in the flat spacetime of special relativity.

The Klein quartic is related to various other surfaces. Geometrically, it is the smallest Hurwitz surface (lowest genus); the next is the Macbeath surface (genus 7), and the following is the First Hurwitz triplet (3 surfaces of genus 14). More generally, it is the most symmetric surface of a given genus (being a Hurwitz surface); in this class, the Bolza surface is the most symmetric genus 2 surface, while Bring's surface is a highly symmetric genus 4 surface – see isometries of Riemann surfaces for further discussion. Algebraically, the (affine) Klein quartic is the modular curve X(7) and the projective Klein quartic is its compactification, just as the dodecahedron (with a cusp in the center of each face) is the modular curve X(5); this explains the relevance for number theory.

For abelian varieties over an algebraically closed field K, the Weil pairing is a nondegenerate pairing :A[n] \times A^\vee[n] \longrightarrow \mu_n for all n prime to the characteristic of K.James Milne, Abelian Varieties, available at www.jmilne.org/math/ Here A^\vee denotes the dual abelian variety of A. This is the so-called Weil pairing for higher dimensions. If A is equipped with a polarisation :\lambda: A \longrightarrow A^\vee, then composition gives a (possibly degenerate) pairing :A[n] \times A[n] \longrightarrow \mu_n. If C is a projective, nonsingular curve of genus ≥ 0 over k, and J its Jacobian, then the theta-divisor of J induces a principal polarisation of J, which in this particular case happens to be an isomorphism (see autoduality of Jacobians).

Because they are closed under addition, subtraction, and multiplication, but not division, the p-adic fractions are a ring but not a field. As a ring, the p-adic fractions are a subring of the rational numbers Q, and an overring of the integers Z. Algebraically, this subring is the localization of the integers Z with respect to the set of powers of p. The set of all p-adic fractions is dense in the real line: any real number x can be arbitrarily closely approximated by dyadic rationals of the form \lfloor 2^i x \rfloor / 2^i. Compared to other dense subsets of the real line, such as the rational numbers, the p-adic rationals are in some sense a relatively "small" dense set, which is why they sometimes occur in proofs.

The inverse of this restriction can be extended uniquely to a ring homomorphism φn from R[X1,...,Xn]Sn to R[X1,...,Xn+1]Sn+1, as follows for instance from the fundamental theorem of symmetric polynomials. Since the images φn(ek(X1,...,Xn)) = ek(X1,...,Xn+1) for k = 1,...,n are still algebraically independent over R, the homomorphism φn is injective and can be viewed as a (somewhat unusual) inclusion of rings; applying φn to a polynomial amounts to adding all monomials containing the new indeterminate obtained by symmetry from monomials already present. The ring ΛR is then the "union" (direct limit) of all these rings subject to these inclusions. Since all φn are compatible with the grading by total degree of the rings involved, ΛR obtains the structure of a graded ring.

In algebraic geometry, the homogeneous coordinate ring R of an algebraic variety V given as a subvariety of projective space of a given dimension N is by definition the quotient ring :R = K[X0, X1, X2, ..., XN] /I where I is the homogeneous ideal defining V, K is the algebraically closed field over which V is defined, and :K[X0, X1, X2, ..., XN] is the polynomial ring in N + 1 variables Xi. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates, for a given choice of basis (in the vector space underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the symmetric algebra.

Another important basic class of examples are representations of polynomial algebras, the free commutative algebras – these form a central object of study in commutative algebra and its geometric counterpart, algebraic geometry. A representation of a polynomial algebra in variables over the field K is concretely a K-vector space with commuting operators, and is often denoted K[T_1,\dots,T_k], meaning the representation of the abstract algebra K[x_1,\dots,x_k] where x_i \mapsto T_i. A basic result about such representations is that, over an algebraically closed field, the representing matrices are simultaneously triangularisable. Even the case of representations of the polynomial algebra in a single variable are of interest – this is denoted by K[T] and is used in understanding the structure of a single linear operator on a finite-dimensional vector space.

If the field F is not algebraically closed, the point of view of function fields is a little more general than that of considering the locus of points, since we include, for instance, "curves" with no points on them. For example, if the base field F is the field R of real numbers, then x2 + y2 = −1 defines an algebraic extension field of R(x), but the corresponding curve considered as a subset of R2 has no points. The equation x2 + y2 = −1 does define an irreducible algebraic curve over R in the scheme sense (an integral, separated one-dimensional schemes of finite type over R). In this sense, the one-to-one correspondence between irreducible algebraic curves over F (up to birational equivalence) and algebraic function fields in one variable over F holds in general. Two curves can be birationally equivalent (i.e.

In number theory, a strong prime is a prime number that is greater than the arithmetic mean of the nearest prime above and below (in other words, it's closer to the following than to the preceding prime). Or to put it algebraically, given a prime number p, where n is its index in the ordered set of prime numbers, . For example, 17 is the seventh prime: the sixth and eighth primes, 13 and 19, add up to 32, and half that is 16; 17 is greater than 16, and so 17 is a strong prime. The first few strong primes are :11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 163, 179, 191, 197, 223, 227, 239, 251, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499 .

In algebra, the Hochster–Roberts theorem, introduced by , states that rings of invariants of linearly reductive groups acting on regular rings are Cohen–Macaulay. In other words, :If V is a rational representation of a linearly reductive group G over a field k, then there exist algebraically independent invariant homogeneous polynomials f_1, \cdots, f_d such that k[V]^G is a free finite graded module over k[f_1, \cdots, f_d]. proved that if a variety over a field of characteristic 0 has rational singularities then so does its quotient by the action of a reductive group; this implies the Hochster–Roberts theorem in characteristic 0 as rational singularities are Cohen–Macaulay. In characteristic p>0, there are examples of groups that are reductive (or even finite) acting on polynomial rings whose rings of invariants are not Cohen–Macaulay.

In differential geometry, representation theory and harmonic analysis, a symmetric space is a pseudo-Riemannian manifold whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (M, g) is said to be symmetric if and only if, for each point p of M, there exists an isometry of M fixing p and acting on the tangent space T_pM as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetries about the endpoints).

In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) C is determined by its Jacobian variety J(C), when the latter is given in the form of a principally polarized abelian variety. In other words, the complex torus J(C), with certain 'markings', is enough to recover C. The same statement holds over any algebraically closed field.James S. Milne, Jacobian Varieties, Theorem 12.1 in From more precise information on the constructed isomorphism of the curves it follows that if the canonically principally polarized Jacobian varieties of curves of genus \geq 2 are k-isomorphic for k any perfect field, so are the curves.James S. Milne, Jacobian Varieties, Corollary 12.2 in This result has had many important extensions.

The problem originally arose in algebraic invariant theory. Here the ring R is given as a (suitably defined) ring of polynomial invariants of a linear algebraic group over a field k acting algebraically on a polynomial ring k[x1, ..., xn] (or more generally, on a finitely generated algebra defined over a field). In this situation the field K is the field of rational functions (quotients of polynomials) in the variables xi which are invariant under the given action of the algebraic group, the ring R is the ring of polynomials which are invariant under the action. A classical example in nineteenth century was the extensive study (in particular by Cayley, Sylvester, Clebsch, Paul Gordan and also Hilbert) of invariants of binary forms in two variables with the natural action of the special linear group SL2(k) on it.

Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic curves; see below.) An elliptic curve is an abelian variety – that is, it has a group law defined algebraically, with respect to which it is an abelian group – and O serves as the identity element. If y2 = P(x), where P is any polynomial of degree three in x with no repeated roots, the solution set is a nonsingular plane curve of genus one, an elliptic curve. If P has degree four and is square-free this equation again describes a plane curve of genus one; however, it has no natural choice of identity element.

At the end of Book 2 of Nova Scientia, Tartaglia proposes to find the length of that initial rectilinear path for a projectile fired at an elevation of 45°, engaging in a Euclidean-style argument, but one with numbers attached to line segments and areas, and eventually proceeds algebraically to find the desired quantity (procederemo per algebra in his words).See Valleriani, Matteo, Metallurgy, Ballistics and Epistemic Instruments: The Nova Scientia of Nicolò Tartaglia, 2013, pp. 176-177. Mary J. Henninger-Voss notes that "Tartaglia's work on military science had an enormous circulation throughout Europe", being a reference for common gunners into the eighteenth century, sometimes through unattributed translations. He influenced Galileo as well, who owned "richly annotated" copies of his works on ballistics as he set about solving the projectile problem once and for all.

When K is the field of complex numbers, this shows that the algebra K[G] is a product of several copies of complex matrix algebras, one for each irreducible representation.The number of the summands can be computed, and turns out to be equal to the number of the conjugacy classes of the group. If the field K has characteristic zero, but is not algebraically closed, for example, K is a field of real or rational numbers, then a somewhat more complicated statement holds: the group algebra K[G] is a product of matrix algebras over division rings over K. The summands correspond to irreducible representations of G over K.One must be careful, since a representation may decompose differently over different fields: a representation may be irreducible over the real numbers but not over the complex numbers.

A geometric solution to this is to intersect the curve not with itself, but with a slightly pushed off version of itself. In the plane, this just means translating the curve in some direction, but in general one talks about taking a curve that is linearly equivalent to , and counting the intersection , thus obtaining an intersection number, denoted . Note that unlike for distinct curves and , the actual points of intersection are not defined, because they depend on a choice of , but the “self intersection points of can be interpreted as generic points on , where . More properly, the self- intersection point of is the generic point of , taken with multiplicity . Alternatively, one can “solve” (or motivate) this problem algebraically by dualizing, and looking at the class of – this both gives a number, and raises the question of a geometric interpretation.

A set of matrices A_1, \ldots, A_k are said to be ' if there is a basis under which they are all upper triangular; equivalently, if they are upper triangularizable by a single similarity matrix P. Such a set of matrices is more easily understood by considering the algebra of matrices it generates, namely all polynomials in the A_i, denoted K[A_1,\ldots,A_k]. Simultaneous triangularizability means that this algebra is conjugate into the Lie subalgebra of upper triangular matrices, and is equivalent to this algebra being a Lie subalgebra of a Borel subalgebra. The basic result is that (over an algebraically closed field), the commuting matrices A,B or more generally A_1,\ldots,A_k are simultaneously triangularizable. This can be proven by first showing that commuting matrices have a common eigenvector, and then inducting on dimension as before.

If K is given inside an algebraically closed field C, then the conjugates can be taken inside C. If no such C is specified, one can take the conjugates in some relatively small field L. The smallest possible choice for L is to take a splitting field over K of pK,α, containing α. If L is any normal extension of K containing α, then by definition it already contains such a splitting field. Given then a normal extension L of K, with automorphism group Aut(L/K) = G, and containing α, any element g(α) for g in G will be a conjugate of α, since the automorphism g sends roots of p to roots of p. Conversely any conjugate β of α is of this form: in other words, G acts transitively on the conjugates.

A rationally connected variety (or uniruled variety) V is a projective algebraic variety over an algebraically closed field such that through every two points there passes the image of a regular map from the projective line into V. Equivalently, a variety is rationally connected if every two points are connected by a rational curve contained in the variety.. This definition differs form that of path connectedness only by the nature of the path, but is very different, as the only algebraic curves which are rationally connected are the rational ones. Every rational variety, including the projective spaces, is rationally connected, but the converse is false. The class of the rationally connected varieties is thus a generalization of the class of the rational varieties. Unirational varieties are rationally connected, but it is not known if the converse holds.

The projective line is a fundamental example of an algebraic curve. From the point of view of algebraic geometry, P1(K) is a non-singular curve of genus 0. If K is algebraically closed, it is the unique such curve over K, up to rational equivalence. In general a (non-singular) curve of genus 0 is rationally equivalent over K to a conic C, which is itself birationally equivalent to projective line if and only if C has a point defined over K; geometrically such a point P can be used as origin to make explicit the birational equivalence.. The function field of the projective line is the field K(T) of rational functions over K, in a single indeterminate T. The field automorphisms of K(T) over K are precisely the group PGL2(K) discussed above.

A quasivariety defined logically as the class of models of a universal Horn theory can equivalently be defined algebraically as a class of structures closed under isomorphisms, subalgebras, and reduced products. Since the notion of reduced product is more intricate than that of direct product, it is sometimes useful to blend the logical and algebraic characterizations in terms of pseudoelementary classes. One such blended definition characterizes a quasivariety as a pseudoelementary class closed under isomorphisms, subalgebras, and direct products (the pseudoelementary property allows "reduced" to be simplified to "direct"). A corollary of this characterization is that one can (nonconstructively) prove the existence of a universal Horn axiomatization of a class by first axiomatizing some expansion of the structure with auxiliary sorts and relations and then showing that the pseudoelementary class obtained by dropping the auxiliary constructs is closed under subalgebras and direct products.

For C of genus 0 there is one such divisor class, namely the class of -P, where P is any point on the curve. In case of higher genus g, assuming the field over which C is defined does not have characteristic 2, the theta characteristics can be counted as :22g in number if the base field is algebraically closed. This comes about because the solutions of the equation on the divisor class level will form a single coset of the solutions of :2D = 0. In other words, with K the canonical class and Θ any given solution of :2Θ = K, any other solution will be of form :Θ \+ D. This reduces counting the theta characteristics to finding the 2-rank of the Jacobian variety J(C) of C. In the complex case, again, the result follows since J(C) is a complex torus of dimension 2g.

Fields of characteristic zero have the most familiar properties; for practical purposes they resemble subfields of the complex numbers (unless they have very large cardinality, that is; in fact, any field of characteristic zero and cardinality at most continuum is (ring-)isomorphic to a subfield of complex numbers).. Enderton states this result explicitly only for algebraically closed fields, but also describes a decomposition of any field as an algebraic extension of a transcendental extension of its prime field, from which the result follows immediately. The p-adic fields or any finite extension of them are characteristic zero fields, much applied in number theory, that are constructed from rings of characteristic pk, as k → ∞. For any ordered field, as the field of rational numbers Q or the field of real numbers R, the characteristic is 0. Thus, number fields and the field of complex numbers C are of characteristic zero.

From now on, we consider that the homogeneous polynomials P_1,\ldots,P_n of degrees d_1,\ldots,d_n have their coefficients in a field , that is that they belong to k[x_1,\dots,x_n]. Their resultant is defined as the element of obtained by replacing in the generic resultant the indeterminate coefficients by the actual coefficients of the P_i. The main property of the resultant is that it is zero if and only if P_1,\ldots,P_n have a nonzero common zero in an algebraically closed extension of . The "only if" part of this theorem results from the last property of the preceding paragraph, and is an effective version of Projective Nullstellensatz: If the resultant is nonzero, then :\langle x_1,\ldots, x_n\rangle^D \subseteq \langle P_1,\ldots,P_n\rangle, where D=d_1+\cdots +d_n-n+1 is the Macaulay degree, and \langle x_1,\ldots, x_n\rangle is the maximal homogeneous ideal.

Alexander Grothendieck's work during the "Golden Age" period at the IHÉS established several unifying themes in algebraic geometry, number theory, topology, category theory and complex analysis. His first (pre-IHÉS) discovery in algebraic geometry was the Grothendieck–Hirzebruch–Riemann–Roch theorem, a generalisation of the Hirzebruch–Riemann–Roch theorem proved algebraically; in this context he also introduced K-theory. Then, following the programme he outlined in his talk at the 1958 International Congress of Mathematicians, he introduced the theory of schemes, developing it in detail in his Éléments de géométrie algébrique (EGA) and providing the new more flexible and general foundations for algebraic geometry that has been adopted in the field since that time. He went on to introduce the étale cohomology theory of schemes, providing the key tools for proving the Weil conjectures, as well as crystalline cohomology and algebraic de Rham cohomology to complement it.

This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality. Diophantine geometry in general is the study of algebraic varieties V over fields K that are finitely generated over their prime fields—including as of special interest number fields and finite fields—and over local fields. Of those, only the complex numbers are algebraically closed; over any other K the existence of points of V with coordinates in K is something to be proved and studied as an extra topic, even knowing the geometry of V. Arithmetic geometry can be more generally defined as the study of schemes of finite type over the spectrum of the ring of integers.

In mathematics, an algebraic variety V in projective space is a complete intersection if the ideal of V is generated by exactly codim V elements. That is, if V has dimension m and lies in projective space Pn, there should exist n − m homogeneous polynomials :Fi(X0, ..., Xn), 1 ≤ i ≤ n − m, in the homogeneous coordinates Xj, which generate all other homogeneous polynomials that vanish on V. Geometrically, each Fi defines a hypersurface; the intersection of these hypersurfaces should be V. The intersection of n-m hypersurfaces will always have dimension at least m, assuming that the field of scalars is an algebraically closed field such as the complex numbers. The question is essentially, can we get the dimension down to m, with no extra points in the intersection? This condition is fairly hard to check as soon as the codimension n − m ≥ 2.

In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras. It states that if G is a connected and solvable linear algebraic group defined over an algebraically closed field and :\rho\colon G \to GL(V) a representation on a nonzero finite-dimensional vector space V, then there is a one-dimensional linear subspace L of V such that : \rho(G)(L) = L. That is, ρ(G) has an invariant line L, on which G therefore acts through a one- dimensional representation. This is equivalent to the statement that V contains a nonzero vector v that is a common (simultaneous) eigenvector for all \rho(g), \,\, g \in G . It follows directly that every irreducible finite-dimensional representation of a connected and solvable linear algebraic group G has dimension one.

In mathematics, the Weil reciprocity law is a result of André Weil holding in the function field K(C) of an algebraic curve C over an algebraically closed field K. Given functions f and g in K(C), i.e. rational functions on C, then :f((g)) = g((f)) where the notation has this meaning: (h) is the divisor of the function h, or in other words the formal sum of its zeroes and poles counted with multiplicity; and a function applied to a formal sum means the product (with multiplicities, poles counting as a negative multiplicity) of the values of the function at the points of the divisor. With this definition there must be the side-condition, that the divisors of f and g have disjoint support (which can be removed). In the case of the projective line, this can be proved by manipulations with the resultant of polynomials.

The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of the variety. In the case of an algebraic variety over the complex numbers, the Zariski topology is thus coarser than the usual topology, as every algebraic set is closed for the usual topology. The generalization of the Zariski topology to the set of prime ideals of a commutative ring follows from Hilbert's Nullstellensatz, that establishes a bijective correspondence between the points of an affine variety defined over an algebraically closed field and the maximal ideals of the ring of its regular functions. This suggests defining the Zariski topology on the set of the maximal ideals of a commutative ring as the topology such that a set of maximal ideals is closed if and only if it is the set of all maximal ideals that contain a given ideal.

A rational curve, also called a unicursal curve, is any curve which is birationally equivalent to a line, which we may take to be a projective line; accordingly, we may identify the function field of the curve with the field of rational functions in one indeterminate F(x). If F is algebraically closed, this is equivalent to a curve of genus zero; however, the field of all real algebraic functions defined on the real algebraic variety x2+y2 = −1 is a field of genus zero which is not a rational function field. Concretely, a rational curve embedded in an affine space of dimension n over F can be parameterized (except for isolated exceptional points) by means of n rational functions of a single parameter t; by reducing these rational functions to the same denominator, the n+1 resulting polynomials define a polynomial parametrization of the projective completion of the curve in the projective space. An example is the rational normal curve, where all these polynomials are monomials.

In abstract algebra, Abhyankar's conjecture is a 1957 conjecture of Shreeram Abhyankar, on the Galois groups of algebraic function fields of characteristic p.. The soluble case was solved by Serre in 1990 and the full conjecture was proved in 1994 by work of Michel Raynaud and David Harbater... The problem involves a finite group G, a prime number p, and the function field K(C) of a nonsingular integral algebraic curve C defined over an algebraically closed field K of characteristic p. The question addresses the existence of a Galois extension L of K(C), with G as Galois group, and with specified ramification. From a geometric point of view, L corresponds to another curve C′, together with a morphism :π : C′ → C. Geometrically, the assertion that π is ramified at a finite set S of points on C means that π restricted to the complement of S in C is an étale morphism. This is in analogy with the case of Riemann surfaces.

For points in the plane or on an algebraic curve, the notion of general position is made algebraically precise by the notion of a regular divisor, and is measured by the vanishing of the higher sheaf cohomology groups of the associated line bundle (formally, invertible sheaf). As the terminology reflects, this is significantly more technical than the intuitive geometric picture, similar to how a formal definition of intersection number requires sophisticated algebra. This definition generalizes in higher dimensions to hypersurfaces (codimension 1 subvarieties), rather than to sets of points, and regular divisors are contrasted with superabundant divisors, as discussed in the Riemann–Roch theorem for surfaces. Note that not all points in general position are projectively equivalent, which is a much stronger condition; for example, any k distinct points in the line are in general position, but projective transformations are only 3-transitive, with the invariant of 4 points being the cross ratio.

An advantage of the Wald test over the other two is that it only requires the estimation of the unrestricted model, which lowers the computational burden as compared to the likelihood- ratio test. However, a major disadvantage is that (in finite samples) it is not invariant to changes in the representation of the null hypothesis; in other words, algebraically equivalent expressions of non-linear parameter restriction can lead to different values of the test statistic. That is because the Wald statistic is derived from a Taylor expansion,, and different ways of writing equivalent nonlinear expressions lead to nontrivial differences in the corresponding Taylor coefficients. Another aberration, known as the Hauck–Donner effect, can occur in binomial models when the estimated (unconstrained) parameter is close to the boundary of the parameter space—for instance a fitted probability being extremely close to zero or one—which results in the Wald test no longer monotonically increasing in the distance between the unconstrained and constraint parameter.

In mathematics, Diophantine geometry is the study of points of algebraic varieties with coordinates in the integers, rational numbers, and their generalizations. These generalizations typically are fields that are not algebraically closed, such as number fields, finite fields, function fields, and p-adic fields (but not the real numbers which are used in real algebraic geometry). It is a sub-branch of arithmetic geometry and is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry. A single equation defines a hypersurface, and simultaneous Diophantine equations give rise to a general algebraic variety V over K; the typical question is about the nature of the set V(K) of points on V with co-ordinates in K, and by means of height functions quantitative questions about the "size" of these solutions may be posed, as well as the qualitative issues of whether any points exist, and if so whether there are an infinite number.

A dodecagonal number is a figurate number that represents a dodecagon. The dodecagonal number for n is given by the formula :5n^2 - 4n; n > 0 The first few dodecagonal numbers are: :1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, 672, 793, 924, 1065, 1216, 1377, 1548, 1729, 1920, 2121, 2332, 2553, 2784, 3025, 3276, 3537, 3808, 4089, 4380, 4681, 4992, 5313, 5644, 5985, 6336, 6697, 7068, 7449, 7840, 8241, 8652, 9073, 9504, 9945 ... The dodecagonal number for n can also be calculated by adding the square of n to four times the (n - 1)th pronic number, or to put it algebraically, D_n = n^2 + 4(n^2 - n). Dodecagonal numbers consistently alternate parity, and in base 10, their units place digits follow the pattern 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. By the Fermat polygonal number theorem, every number is the sum of at most 12 dodecagonal numbers.

While local time could explain the negative aether drift experiments to first order to v/c, it was necessary – due to other unsuccessful aether drift experiments like the Trouton–Noble experiment – to modify the hypothesis to include second-order effects. The mathematical tool for that is the so-called Lorentz transformation. Voigt in 1887 had already derived a similar set of equations (although with a different scale factor). Afterwards, Larmor in 1897 and Lorentz in 1899 derived equations in a form algebraically equivalent to those which are used up to this day, although Lorentz used an undetermined factor l in his transformation. In his paper Electromagnetic phenomena in a system moving with any velocity smaller than that of light (1904) Lorentz attempted to create such a theory, according to which all forces between the molecules are affected by the Lorentz transformation (in which Lorentz set the factor l to unity) in the same manner as electrostatic forces.

As complex numbers use an imaginary unit to complement the real line, Hamilton considered the vector to be the imaginary part of a quaternion: :The algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion.W. R. Hamilton (1846) London, Edinburgh & Dublin Philosophical Magazine 3rd series 29 27 Several other mathematicians developed vector-like systems in the middle of the nineteenth century, including Augustin Cauchy, Hermann Grassmann, August Möbius, Comte de Saint-Venant, and Matthew O'Brien. Grassmann's 1840 work Theorie der Ebbe und Flut (Theory of the Ebb and Flow) was the first system of spatial analysis that is similar to today's system, and had ideas corresponding to the cross product, scalar product and vector differentiation. Grassmann's work was largely neglected until the 1870s.

Amongst his notable discoveries are the Hitchin-Thorpe inequality; Hitchin's projectively flat connection over Teichmüller space; the Atiyah–Hitchin monopole metric; the Atiyah–Hitchin–Singer theorem; the ADHM construction of instantons (of Michael Atiyah, Vladimir Drinfeld, Hitchin, and Yuri Manin); the hyperkähler quotient (of Hitchin, Anders Karlhede, Ulf Lindström and Martin Roček); Higgs bundles, which arise as solutions to the Hitchin equations, a 2-dimensional reduction of the self-dual Yang–Mills equations; and the Hitchin system, an algebraically completely integrable Hamiltonian system associated to the data of an algebraic curve and a complex reductive group. He and Shoshichi Kobayashi independently conjectured the Kobayashi–Hitchin correspondence. Higgs bundles, which are also developed in the work of Carlos Simpson, are closely related to the Hitchin system, which has an interpretation as a moduli space of semistable Higgs bundles over a compact Riemann surface or algebraic curve. This moduli space has emerged as a focal point for deep connections between algebraic geometry, differential geometry, hyperkähler geometry, mathematical physics, and representation theory.

Suppose that F is an integral domain, such as the field Q(x1,...,xn) of rational functions in n variables over the rational numbers Q. A cluster of rank n consists of a set of n elements {x, y, ...} of F, usually assumed to be an algebraically independent set of generators of a field extension F. A seed consists of a cluster {x, y, ...} of F, together with an exchange matrix B with integer entries bx,y indexed by pairs of elements x, y of the cluster. The matrix is sometimes assumed to be skew-symmetric, so that bx,y = –by,x for all x and y. More generally the matrix might be skew-symmetrizable, meaning there are positive integers dx associated with the elements of the cluster such that dxbx,y = –dyby,x for all x and y. It is common to picture a seed as a quiver with vertices the generating set, by drawing bx,y arrows from x to y if this number is positive.

All the reflections are conjugate to each other in case n is odd, but they fall into two conjugacy classes if n is even. If we think of the isometries of a regular n-gon: for odd n there are rotations in the group between every pair of mirrors, while for even n only half of the mirrors can be reached from one by these rotations. Geometrically, in an odd polygon every axis of symmetry passes through a vertex and a side, while in an even polygon there are two sets of axes, each corresponding to a conjugacy class: those that pass through two vertices and those that pass through two sides. Algebraically, this is an instance of the conjugate Sylow theorem (for n odd): for n odd, each reflection, together with the identity, form a subgroup of order 2, which is a Sylow 2-subgroup ( is the maximum power of 2 dividing ), while for n even, these order 2 subgroups are not Sylow subgroups because 4 (a higher power of 2) divides the order of the group.

The second-order QSS method, QSS2, follows the same principle as QSS1, except that it defines q(t) as a piecewise linear approximation of the trajectory x(t) that updates its trajectory as soon as the two differ from each other by one quantum. The pattern continues for higher-order approximations, which define the quantized state q(t) as successively higher-order polynomial approximations of the system's state. It is important to note that, while in principle a QSS method of arbitrary order can be used to model a continuous-time system, it is seldom desirable to use methods of order higher than four, as the Abel–Ruffini theorem implies that the time of the next quantization, t, cannot (in general) be explicitly solved for algebraically when the polynomial approximation is of degree greater than four, and hence must be approximated iteratively using a root-finding algorithm. In practice, QSS2 or QSS3 proves sufficient for many problems and the use of higher-order methods results in little, if any, additional benefit.

Given a field k, and an algebraically closed extension K of k, an affine variety X over k is the set of common zeros in K^n of a collection of polynomials with coefficients in k: :f_1(x_1,\ldots,x_n)=0,\ldots, f_r(x_1,\dots,x_n)=0. These common zeros are called the points of X. A k-rational point (or k-point) of X is a point of X that belongs to kn, that is, a sequence (a1,...,an) of n elements of k such that fj(a1,...,an) = 0 for all j. The set of k-rational points of X is often denoted X(k). Sometimes, when the field k is understood, or when k is the field Q of rational numbers, one says "rational point" instead of "k-rational point". For example, the rational points of the unit circle of equation :x^2+y^2=1 are the pairs of rational numbers :\left(\frac ac, \frac bc\right), where (a, b, c) is a Pythagorean triple.

This agrees with the previous definitions when X is an affine or projective variety (viewed as a scheme over k). When X is a variety over an algebraically closed field k, much of the structure of X is determined by its set X(k) of k-rational points. For a general field k, however, X(k) gives only partial information about X. In particular, for a variety X over a field k and any field extension E of k, X also determines the set X(E) of E-rational points of X, meaning the set of solutions of the equations defining X with values in E. Example: Let X be the conic curve x2 \+ y2 = −1 in the affine plane A2 over the real numbers R. Then the set of real points X(R) is empty, because the square of any real number is nonnegative. On the other hand, in the terminology of algebraic geometry, the algebraic variety X over R is not empty, because the set of complex points X(C) is not empty.

To determine whether a given line bundle on a projective variety X is ample, the following numerical criteria (in terms of intersection numbers) are often the most useful. It is equivalent to ask when a Cartier divisor D on X is ample, meaning that the associated line bundle O(D) is ample. The intersection number D\cdot C can be defined as the degree of the line bundle O(D) restricted to C. In the other direction, for a line bundle L on a projective variety, the first Chern class c_1(L) means the associated Cartier divisor (defined up to linear equivalence), the divisor of any nonzero rational section of L. On a smooth projective curve X over an algebraically closed field k, a line bundle L is very ample if and only if h^0(X,L\otimes O(-x-y))=h^0(X,L)-2 for all k-rational points x,y in X.Hartshorne (1977), Proposition IV.3.1. Let g be the genus of X. By the Riemann–Roch theorem, every line bundle of degree at least 2g + 1 satisfies this condition and hence is very ample.

Another is defined the same way, but using split octonions instead of octonions. The final is constructed from the non-split octonions using a different standard involution. Over any algebraically closed field, there is just one Albert algebra, and its automorphism group G is the simple split group of type F4.Springer & Veldkamp (2000) 7.2 (For example, the complexifications of the three Albert algebras over the real numbers are isomorphic Albert algebras over the complex numbers.) Because of this, for a general field F, the Albert algebras are classified by the Galois cohomology group H1(F,G).Knus et al (1998) p.517 The Kantor–Koecher–Tits construction applied to an Albert algebra gives a form of the E7 Lie algebra. The split Albert algebra is used in a construction of a 56-dimensional structurable algebra whose automorphism group has identity component the simply-connected algebraic group of type E6. The space of cohomological invariants of Albert algebras a field F (of characteristic not 2) with coefficients in Z/2Z is a free module over the cohomology ring of F with a basis 1, f3, f5, of degrees 0, 3, 5.

The reason of the interest for Diophantine equations, in the elliptic curve case, is that K may not be algebraically closed. There can exist curves C that have no point defined over K, and which become isomorphic over a larger field to E, which by definition has a point over K to serve as identity element for its addition law. That is, for this case we should distinguish C that have genus 1, from elliptic curves E that have a K-point (or, in other words, provide a Diophantine equation that has a solution in K). The curves C turn out to be torsors over E, and form a set carrying a rich structure in the case that K is a number field (the theory of the Selmer group). In fact a typical plane cubic curve C over Q has no particular reason to have a rational point; the standard Weierstrass model always does, namely the point at infinity, but you need a point over K to put C into that form over K. This theory has been developed with great attention to local analysis, leading to the definition of the Tate-Shafarevich group.

Since the second Betti number of a K3 surface is always 22, this property means that the surface has 22 independent elements in its Picard group (ρ = 22). From what we have said, a K3 surface with Picard number 22 must be supersingular. Conversely, the Tate conjecture would imply that every supersingular K3 surface over an algebraically closed field has Picard number 22. This is now known in every characteristic p except 2, since the Tate conjecture was proved for all K3 surfaces in characteristic p at least 3 by Nygaard-Ogus (1985), , , and . To see that K3 surfaces with Picard number 22 exist only in positive characteristic, one can use Hodge theory to prove that the Picard number of a K3 surface in characteristic zero is at most 20. In fact the Hodge diamond for any complex K3 surface is the same (see classification), and the middle row reads 1, 20, 1. In other words, h2,0 and h0,2 both take the value 1, with h1,1 = 20. Therefore, the dimension of the space spanned by the algebraic cycles is at most 20 in characteristic zero; surfaces with this maximum value are sometimes called singular K3 surfaces.

Manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low- dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such as exotic differentiable structures on R4. Thus the topological classification of 4-manifolds is in principle easy, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the generalized Poincaré conjecture; see Gluck twists. The distinction is because surgery theory works in dimension 5 and above (in fact, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above is controlled algebraically by surgery theory. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work, and other phenomena occur.

Take two smooth curves C and D in a smooth projective 3-fold P, intersecting in two points c and d that are nodes for the reducible curve C\cup D. For some applications these should be chosen so that there is a fixed-point-free automorphism exchanging the curves C and D and also exchanging the points c and d. Hironaka's example V is obtained by blowing up the curves C and D, with C blown up first at the point c and D blown up first at the point d. Then V has two smooth rational curves L and M lying over c and d such that L+M is algebraically equivalent to 0, so V cannot be projective. For an explicit example of this configuration, take t to be a point of order 2 in an elliptic curve E, take P to be E\times E/(t)\times E/(t), take C and D to be the sets of points of the form (x,x,0) and (x,0,x), so that c and d are the points (0,0,0) and (t,0,0), and take the involution σ to be the one taking (x,y,z) to (x+t,z,y).