Algebraically closed field

A field $k$ is algebraically closed if every nonconstant polynomial $f(x)\in k[x]$ has a root in $k$ . Equivalently, every nonconstant polynomial factors completely into linear factors:

f(x)=c\prod_{i=1}^n(x-a_i), \qquad c,a_i\in k,

or, equivalently, $k$ has no proper finite algebraic field extension. An algebraic closure of a field $F$ is an algebraic extension $\overline F/F$ whose field $\overline F$ is algebraically closed.

Examples

Standard examples include $\mathbb C$ and $\overline{\mathbb F}_p$ ; neither $\mathbb R$ nor a finite field is algebraically closed.

Remarks

Algebraic closedness concerns polynomial equations, not topological closedness or metric completeness.