Algebraic closure

Using Zorn's Lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure is unique in the following sense: if L and M are algebraic closures of a field K, then there is an isomorphism f : L -> M such that f(x) = x for each x in 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 which 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.

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.

• 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 transcendantal extensions of the rational 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).