Uniqueness of Algebraic Closures

Theorem (Uniqueness up to K-isomorphism).
Let $K$ be a field and let $\overline{K}_1$ and $\overline{K}_2$ be algebraic closures of $K$. Then there exists a $K$-isomorphism

fixing $K$ pointwise. The isomorphism is not canonical.

This complements existence of algebraic closures .

Examples.

1. Any two choices of $\overline{\mathbb{Q}}$ (inside possibly different ambient fields) are isomorphic over $\mathbb{Q}$.
2. Any algebraic closure of $\mathbb{F}_p$ is isomorphic (over $\mathbb{F}_p$) to $\bigcup_{n\ge1}\mathbb{F}_{p^n}$.
3. If $\overline{K}$ is fixed, any other algebraic closure can be identified with a $K$-subfield of $\overline{K}$ via such an isomorphism (after choosing one).

Related. Zorn's lemma , axiom of choice .