# 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.

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 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]}

It can be shown along the same lines that for any subset *S* of *K*[*x*], there exists a splitting field of *S* over *K*.

An algebraic closure *K ^{alg}* of

*K*contains a unique separable extension

*K*of

^{sep}*K*containing all (algebraic) separable extensions of

*K*within

*K*. This subextension is called a

^{alg}**separable closure**of

*K*. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of

*K*, of degree > 1. Saying this another way,

^{sep}*K*is contained in a

*separably-closed*algebraic extension field. It is unique (up to isomorphism).

^{[7]}

In general, the absolute Galois group of *K* is the Galois group of *K ^{sep}* over

*K*.

^{[8]}