Existence of Algebraic Closures

Theorem (Existence).
For every field $K$, there exists an extension field $\overline{K}$ such that:

1. $\overline{K}/K$ is an algebraic extension , and
2. $\overline{K}$ is algebraically closed .

Such a field $\overline{K}$ is called an algebraic closure of $K$. Standard proofs use Zorn's lemma (hence choice ).

Examples.

1. An algebraic closure of $\mathbb{R}$ is $\mathbb{C}$ (every complex number satisfies a quadratic over $\mathbb{R}$).
2. An algebraic closure of $\mathbb{F}_p$ is the union $\bigcup_{n\ge1}\mathbb{F}_{p^n}$ inside a fixed ambient algebraic closure.
3. $\overline{\mathbb{Q}}$ denotes an algebraic closure of $\mathbb{Q}$ (often viewed inside $\mathbb{C}$).

Related. uniqueness of algebraic closures .