Perfect closure

Summary

The perfect closure of a field $K$ of characteristic $p > 0$ is the smallest perfect field that contains $K$ inside an algebraic closure. It is obtained by adjoining all $p^n$-th roots of all elements of $K$, so it is a purely inseparable extension of $K$. It is the purely inseparable counterpart of the separable closure. For a perfect field, and for any field of characteristic $0$, the perfect closure is the field itself.