Algebraic closure

In mathematics, an algebraic closure of a field K is an algebraic extension of K which is algebraically closed.

Every field has an algebraic closure, and it is unique up to isomorphism. The proof of the existence of algebraic closures requires the axiom of choice in the form of Zorn's lemma.

Examples:

• The Fundamental Theorem of Algebra states that the algebraic closure of the field of real numbers is the field of complex numbers.

• The algebraic closure of the field of rational numbers is the field of algebraic numbers.

• For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order pⁿ for each positive integer n (and is in fact the union of these copies).