In mathematics, particularly abstract algebra, an algebraic closure of a fieldK is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemma or the weaker ultrafilter 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. 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 overK 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.
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.
For a finite field of prime power order q, the algebraic closure is a countably infinite field that contains a copy of the field of order qn for each positive integern.
Existence of an algebraic closure and splitting fields
Let be the set of all monic irreducible polynomials in K. For each, introduce new variables where. Let R be the polynomial ring over K generated by for all and all. Write with. Let I be the ideal in R generated by the. Since I is strictly smaller thanR, Zorn's lemma implies that there exists a maximal idealM in R that contains I. The field K1=R/M has the property that every polynomial with coefficients in K splits as the product of and hence has all roots in K1. In the same way, an extension K2 of K1 can be constructed, etc. The union of all these extensions is the algebraic closure of K, because any polynomial with coefficients in this new field has its coefficients in some Kn with sufficiently largen, and then its roots are in Kn+1, and hence in the union itself. It can be shown along the same lines that for any subset S of K, there exists a splitting field of S over K.
Separable closure
An algebraic closure Kalg of K contains a unique separable extensionKsep of K containing all separable extensions of K within Kalg. This subextension is called a separable closure of K. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of Ksep, of degree > 1. Saying this another way, K is contained in a separably-closed algebraic extension field. It is unique. The separable closure is the full algebraic closure if and only ifK is a perfect field. For example, if K is a field of characteristic p and if X is transcendental over K, is a non-separable algebraic field extension. In general, the absolute Galois group of K is the Galois group of Ksep over K. For example, if, then its separable closure is
