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, it can be shown that every field has an algebraic closure,^{[1]}^{[2]}^{[3]} 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.^{[3]}
Examples

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 infinite field that contains a copy of the field of order q^{n} for each positive integer n (and is in fact the union of these copies).^{[4]}
Existence of an algebraic closure and splitting fields
Let S = \{ f_{\lambda}  \lambda \in \Lambda\} be the set of all monic irreducible polynomials in K[x]. For each f_{\lambda} \in S, introduce new variables u_{\lambda,1},\ldots,u_{\lambda,d} where d = {\rm degree}(f_{\lambda}). Let R be the polynomial ring over K generated by u_{\lambda,i} for all \lambda \in \Lambda and all i \leq {\rm degree}(f_{\lambda}). Write

f_{\lambda}  \prod_{i=1}^d (xu_{\lambda,i}) = \sum_{j=0}^{d1} r_{\lambda,j} \cdot x^j \in R[x]
with r_{\lambda,j} \in R. Let I be the ideal in R generated by the r_{\lambda,j}. By Zorn's lemma, there exists a maximal ideal M in R that contains I. Now R/M is an algebraic closure of K; every f_{\lambda} splits as the product of the x(u_{\lambda,i} + M).
The same proof also shows that for any subset S of K[x], there exists a splitting field of S over K.
Separable closure
An algebraic closure K^{alg} of K contains a unique separable extension K^{sep} of K containing all (algebraic) separable extensions of K within K^{alg}. 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 K^{sep}, of degree > 1. Saying this another way, K is contained in a separablyclosed algebraic extension field. It is essentially unique (up to isomorphism).^{[5]}
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)(\sqrt[p]{X}) \supset K(X) is a nonseparable algebraic field extension.
In general, the absolute Galois group of K is the Galois group of K^{sep} over K.^{[6]}
See also
References

^ McCarthy (1991) p.21

^ M. F. Atiyah and I. G. Macdonald (1969). Introduction to commutative algebra. AddisonWesley publishing Company. pp. 1112.

^ ^{a} ^{b} Kaplansky (1972) pp.7476

^ .

^ McCarthy (1991) p.22

^
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.