Algebraic closure

Let $F$ be a field .

Definition (algebraic closure). An algebraic closure of $F$ is a field $\overline F$ equipped with an inclusion $F\subseteq \overline F$ such that:

1. $\overline F/F$ is an algebraic extension , i.e. every element of $\overline F$ is algebraic over $F$;
2. $\overline F$ is algebraically closed: every nonconstant polynomial in $\overline F[x]$ splits completely into linear factors over $\overline F$.

In particular, every $f(x)\in F[x]$ splits in $\overline F[x]$, so $\overline F$ contains a splitting field for every polynomial over $F$.

Existence and uniqueness (up to $F$-isomorphism) are treated in existence of algebraic closures and uniqueness of algebraic closures . A useful consequence is that any algebraic field extension $K/F$ admits an $F$-embedding into $\overline F$.

Examples.

1. $\mathbb{C}$ is an algebraic closure of $\mathbb{R}$: it is algebraically closed, and $\mathbb{C}/\mathbb{R}$ is algebraic of degree $2$ (generated by $i$).
2. Inside $\mathbb{C}$, the set $\overline{\mathbb{Q}}$ of algebraic numbers (complex numbers algebraic over $\mathbb{Q}$) forms an algebraic closure of $\mathbb{Q}$.
3. For a finite field $\mathbb{F}_p$, one can realize an algebraic closure as $\overline{\mathbb{F}_p}=\bigcup_{n\ge 1}\mathbb{F}_{p^n}$ inside a fixed large field, where $\mathbb{F}_{p^n}$ ranges over all finite extensions of $\mathbb{F}_p$.