Algebraically closed field

Definition

Passing from $F$ to an algebraically closed field $\overline F$ makes finite separable extensions split into visible geometric points. For example, after a suitable base change, a finite Galois cover becomes a disjoint union indexed by its Galois group.

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:

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.

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