Separably closed field

Summary

A separably closed field is a field with no proper finite separable extension. Equivalently, every separable polynomial over it of positive degree has a root in it. Every algebraically closed field is separably closed, and a field is algebraically closed if and only if it is separably closed and perfect. Separably closed fields have trivial absolute Galois group, and the separable closure of a field is the separably closed field built from it.