Algebraic extension

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If F and G are fields and G contains F, then the field extension G/F is called algebraic if every element of G is algebraic over F, meaning that for every element x of G there exists a non-zero polynomial p with coefficients in F such that p(x) = 0.

For every non-zero polynomial p with coefficients in F, there is an algebraic extension G of F and an x in G such that p(x) = 0. F[x], the set of all polynomials in x with coefficients in F, is a field, and it is unique up to isomorphism if and only if p is irreducible. The isomorphism is not, in general, unique: the group of automorphisms of F[x] over F is called the Galois group of x.

A field with no algebraic extensions is called algebraically closed. Every field is contained in an algebraically closed field, but proving this in general requires some form of the axiom of choice.

Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with variables in M, such that p(x) is true and the set {y in N | p(y)} is finite. It turns out that applying this definition to the theory of fields gives the usual definition of algebraic extension. The Galois group of N over M can again be defined as the group of automorphisms, and it turns out that most of the theory of Galois groups can be developed for the general case.