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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 lemma [ 1 ] [ 2 ] [ 3 ] or the weaker ultrafilter lemma , [ 4 ] [ 5 ] it can be shown that every field has an algebraic closure , and that the ...
Every field is contained in an algebraically closed field , and the roots in of the polynomials with coefficients in form an algebraically closed field called an algebraic closure of . Given two algebraic closures of K {\displaystyle K} there are isomorphisms between them that fix the elements of K . {\displaystyle K.}
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.)
Either k has characteristic 0, or, when k has characteristic p > 0, the Frobenius endomorphism x ↦ x p is an automorphism of k. The separable closure of k is algebraically closed. Every reduced commutative k-algebra A is a separable algebra; i.e., is reduced for every field extension F/k. (see below)
This implies that an extension is algebraic if and only if it is the union of its finite subextensions, and that every finite extension is algebraic. Every field K has an algebraic closure, which is up to an isomorphism the largest extension field of K which is algebraic over K, and also the smallest extension field such that every polynomial ...
Writing K for some algebraic closure of /, the canonical map : / extends to ~:. Since B is a field, ϕ ~ {\displaystyle {\widetilde {\phi }}} is injective and so B is algebraic (thus finite algebraic) over A / m {\displaystyle A/{\mathfrak {m}}} .
If F is an ordered field, 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 identical on F [2] (note that every ring homomorphism between real closed fields automatically is order preserving ...
Krasner's lemma can be used to show that -adic completion and separable closure of global fields commute. [3] In other words, given a prime of a global field L, the separable closure of the -adic completion of L equals the ¯-adic completion of the separable closure of L (where ¯ is a prime of L above ).