Algebraic closure

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. 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 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 over K 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. Let S = { f λ | λ ∈ Λ } {displaystyle S={f_{lambda }|lambda in Lambda }} be the set of all monic irreducible polynomials in K.For each f λ ∈ S {displaystyle f_{lambda }in S} , introduce new variables u λ , 1 , … , u λ , d {displaystyle u_{lambda ,1},ldots ,u_{lambda ,d}} where d = d e g r e e ( f λ ) {displaystyle d={ m {degree}}(f_{lambda })} .Let R be the polynomial ring over K generated by u λ , i {displaystyle u_{lambda ,i}} for all λ ∈ Λ {displaystyle lambda in Lambda } and all i ≤ d e g r e e ( f λ ) {displaystyle ileq { m {degree}}(f_{lambda })} . Write with r λ , j ∈ R {displaystyle r_{lambda ,j}in R} .Let I be the ideal in R generated by the r λ , j {displaystyle r_{lambda ,j}} . Since I is strictly smaller than R,Zorn's lemma implies that there exists a maximal ideal M in R that contains I.The field K1=R/M has the property that every polynomial f λ {displaystyle f_{lambda }} with coefficients in K splits as the product of x − ( u λ , i + M ) , {displaystyle x-(u_{lambda ,i}+M),} 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 large n, 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. An algebraic closure Kalg of K contains a unique separable extension Ksep of K containing all (algebraic) 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 (up to isomorphism). The separable closure is the full algebraic closure if and only if K is a perfect field. For example, if K is a field of characteristic p and if X is transcendental over K, K ( X ) ( X p ) ⊃ K ( X ) {displaystyle K(X)({sqrt{X}})supset K(X)} is a non-separable algebraic field extension.