** Every field has an algebraic closure.

In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.

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.

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

Using Zorn's lemma, 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.

If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

In abstract algebra, a field extension L / K is called algebraic if every element of L is algebraic over K, i. e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i. e. which contain transcendental elements, are called transcendental.

If a is algebraic over K, then K, 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. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

The sum, difference, product and quotient of two algebraic numbers is again algebraic ( this fact can be demonstrated using the resultant ), and the algebraic numbers therefore form a field, sometimes denoted by A ( which may also denote the adele ring ) or < span style =" text-decoration: overline ;"> Q </ span >.

* For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order p < sup > n </ sup > for each positive integer n ( and is in fact the union of these copies ).

The questions range from counting ( e. g., the number of graphs on n vertices with k edges ) to structural ( e. g., which graphs contain Hamiltonian cycles ) to algebraic questions ( e. g., given a graph G and two numbers x and y, does the Tutte polynomial T < sub > G </ sub >( x, y ) have a combinatorial interpretation ?).

Several of these methods may be directly extended to give definitions of e < sup > z </ sup > for complex values of z simply by substituting z in place of x and using the complex algebraic operations.

Three-dimensional right conoid surface plot, described by the elementary algebraic trigonometrical equations < center >,, </ center >

< center > More familiar two-dimensional algebraic equation plot ( red line )</ center >

A field is therefore an algebraic structure 〈 F, +, ·, −, < sup >− 1 </ sup >, 0, 1 〉; of type 〈 2, 2, 1, 1, 0, 0 〉, consisting of two abelian groups:

His first construction shows how to write the real algebraic numbers as a sequence a < sub > 1 </ sub >, a < sub > 2 </ sub >, a < sub > 3 </ sub >, ....

Conversely, given a groupoid G in the algebraic sense, let G < sub > 0 </ sub > be the set of all elements of the form x * x < sup >− 1 </ sup > with x varying through G and define G ( x * x < sup >-1 </ sup >, y * y < sup >-1 </ sup >) as the set of all elements f such that y * y < sup >-1 </ sup > * f * x * x < sup >-1 </ sup > exists.

A recent result of Bary-Soroker shows that for a field K to be Hilbertian it suffices to consider the case of and absolutely irreducible, that is, irreducible in the ring K < sup > alg </ sup >, where K < sup > alg </ sup > is the algebraic closure of K.

This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG ( 2, R ), RP < sup > 2 </ sup >, or P < sub > 2 </ sub >( R ) among other notations.

In particular, this field contains all the numbers named in the mathematical constants article, and all algebraic numbers ( and therefore all rational numbers ).

of a number field K contains an integral ideal of norm not exceeding a certain bound, depending on K, called Minkowski's bound: the finiteness of the class number of an algebraic number field follows immediately.

The ring of algebraic integers O < sub > K </ sub > contains √ since this is a root of the monic polynomial x < sup > 2 </ sup > − d. Moreover, if d ≡ 1 ( mod 4 ) the element ( 1 + √)/ 2 is also an algebraic integer.

The set of algebraic numbers ( ) is the algebraic closure of the rationals, and contains the roots of all polynomials ( including i for instance ).

The term localization originates in algebraic geometry: if R is a ring of functions defined on some geometric object ( algebraic variety ) V, and one wants to study this variety " locally " near a point p, then one considers the set S of all functions which are not zero at p and localizes R with respect to S. The resulting ring R * contains only information about the behavior of V near p. Cf.

Values of algebraic types are analyzed with pattern matching, which identifies a value by its constructor or field names and extracts the data it contains.

coincides with the one predicted by algebraic K-theory, and contains the category of Chow motives in a suitable sense ( and other properties ).

The trace form for a finite degree field extension L / K has non-negative signature for any field ordering of K. The converse, that every Witt equivalence class with non-negative signature contains a trace form, is true for algebraic number fields K.

* Every real algebraic number field K of degree n contains a PV number of degree n. This number is a field generator.

For example an algebraic curve C either meets a given line L in a finite number of points, or possibly contains all of L. Thus a curve intersecting any line in an infinite number of points, while not containing it, must be transcendental.

The book contains specific algebraic explanations for each of the above operations.

If α is a nonnegative integer n, then the ( n + 1 ) th term and all later terms in the series are 0, since each contains a factor ( n − n ); thus in this case the series is finite and gives the algebraic binomial formula.

Let G be a semisimple Lie group or algebraic group over, and fix a maximal torus T along with a Borel subgroup B which contains T. Let λ be an integral weight of T ; λ defines in a natural way a one-dimensional representation C < sub > λ </ sub > of B, by pulling back the representation on T = B / U, where U is the unipotent radical of B.

Similarly, cubic equations with three real solutions have an algebraic solution that is unhelpful in that it contains cube roots of complex numbers ; again an alternative solution exists in terms of trigonometric functions of real terms.

If K is a field and L is an algebraic extension of K, then there is some algebraic extension M of L such that M is a normal extension of K. Furthermore, up to isomorphism there is only one such extension which is minimal, i. e. such that the only subfield of M which contains L and which is a normal extension of K is M itself.

suggests that Sylvester may have been motivated by a related phenomenon in algebraic geometry, in which the inflection points of a cubic curve in the complex projective plane form a configuration of nine points and twelve lines in which each line determined by two of the points contains a third point.

The known pattern to this sequence comes from algebraic number theory, specifically, this sequence is the set of primes p such that 10 is a primitive root modulo p. A conjecture of Emil Artin is that this sequence contains 37. 395 ..% of the primes.

The unipotent radical of an algebraic group G is the set of unipotent elements in the radical of G. It is a connected unipotent normal subgroup of G, and contains all other such subgroups.

* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

This paper not only introduced the Gaussian integers and proved they are a unique factorization domain, it also introduced the terms norm, unit, primary, and associate, which are now standard in algebraic number theory.

In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field ( or more generally any Dedekind domain ) can be described by a certain group known as an ideal class group ( or class group ).

It was shown that while rings of algebraic integers do not always have unique factorization into primes ( because they need not be principal ideal domains ), they do have the property that every proper ideal admits a unique factorization as a product of prime ideals ( that is, every ring of algebraic integers is a Dedekind domain ).

One of the first properties of Z that can fail in the ring of integers O of an algebraic number field K is that of the unique factorization of integers into prime numbers.

When a complex torus carries the structure of an algebraic variety, this structure is necessarily unique.

This is a result of Claude Chevalley: if K is a perfect field, and G an algebraic group over K, there exists a unique normal closed subgroup H in G, such that H is a linear group and G / H an abelian variety.

Every field has an algebraic closure, and it is unique up to an isomorphism that fixes F.

* Every holomorphic vector bundle on a projective variety is induced by a unique algebraic vector bundle.

More specifically, if is an algebraic extension and if, then S is the unique intermediate field that is separable over F and over which E is purely inseparable.

If are two coherent algebraic sheaves on and if is a map of sheaves of modules then there exists a unique map of sheaves of modules with f =

If F is an ordered field ( not just orderable, but a definite ordering is fixed as part of the structure ), the Artin – Schreier theorem states that F has an algebraic extension, called the real closure K of F, such that K is a real closed field whose ordering is an extension of the given ordering on F, and is unique up to a unique isomorphism of fields ( note that every ring homomorphism between real closed fields automatically is order preserving, because x ≤ y if and only if ∃ z y = x + z < sup > 2 </ sup >).

It is a collection of 130 algebraic problems giving numerical solutions of determinate equations ( those with a unique solution ) and indeterminate equations.

Every planar graph has an algebraic dual, which is in general not unique ( any dual defined by a plane embedding will do ).

Similarly to the case of algebraic closure, there is an analogous theorem for real closure: if K is a real closed field, then the field of Puiseux series over K is the real closure of the field of formal Laurent series over K. ( This implies the former theorem since any algebraically closed field of characteristic zero is the unique quadratic extension of some real-closed field.