Using Zorn's 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* 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 transcendental 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*p*^{n}for each positive integer*n*(and is in fact the union of these copies).