Perfect field

Summary

A perfect field is a field over which every irreducible polynomial has distinct roots. Equivalently, it has characteristic $0$, or it has characteristic $p > 0$ and every element is a $p$-th power. Over a perfect field every algebraic extension is separable, so Galois theory applies without restrictions. The fields $\Q$, $\R$, $\C$, all finite fields and all algebraically closed fields are perfect.