Algebraic closure

Definition. An algebraic closure of a field $K$ is a field $\overline{K}$ together with an inclusion $K\subseteq \overline{K}$ such that:

1. $\overline{K}/K$ is an algebraic extension , and
2. $\overline{K}$ is algebraically closed (every nonconstant polynomial in $\overline{K}[x]$ has a root in $\overline{K}$).

Existence and uniqueness (up to $K$-isomorphism) are addressed in existence and uniqueness .

See also. splitting field , normal extension .

Examples.

1. $\mathbb{C}$ is an algebraic closure of $\mathbb{R}$ (since $[\mathbb{C}:\mathbb{R}]=2$ and $\mathbb{C}$ is algebraically closed).
2. The field $\overline{\mathbb{Q}}$ of all algebraic numbers is an algebraic closure of $\mathbb{Q}$.
3. An algebraic closure of $\mathbb{F}_p$ can be realized as $\bigcup_{n\ge 1}\mathbb{F}_{p^n}$.