Separable closure

Summary

The separable closure of $K$ in an algebraic extension $L$ is the set of elements of $L$ that are separable over $K$. It is the largest separable subextension of $L/K$. The separable closure of a field $K$, written $K^{\mathrm{sep}}$, is the separable closure of $K$ in an algebraic closure; it is a separably closed field. Its automorphism group over $K$ is the absolute Galois group of $K$. For a perfect field the separable closure is the whole algebraic closure.