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There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers, e.g. the algebraic closure of Q(π). For a finite field of prime power order q, the algebraic closure is a countably ...
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.
When F is the algebraic closure of a finite field, the result follows from Hilbert's Nullstellensatz. The Ax–Grothendieck theorem for complex numbers can therefore be proven by showing that a counterexample over C would translate into a counterexample in some algebraic extension of a finite field.
Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group. (When K is a perfect field, K sep is the same as an algebraic closure K alg of K. This holds e.g. for K of characteristic zero, or K a finite field.)
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules.
This equality implies that, if [E : F] is finite, and U is an intermediate field between F and E, then [E : F] sep = [E : U] sep ⋅[U : F] sep. [20] The separable closure F sep of a field F is the separable closure of F in an algebraic closure of F. It is the maximal Galois extension of F.
Algebraic closure An algebraic closure of a field F is an algebraic extension of F which is algebraically closed. Every field has an algebraic closure, and it is unique up to an isomorphism that fixes F. Transcendental Those elements of an extension field of F that are not algebraic over F are transcendental over F. Algebraically independent ...
A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension. [2] The property of an extension being Galois behaves well with respect to field composition and intersection. [3]