Purely inseparable extension

Summary

A purely inseparable extension is an algebraic extension in which every element has some $p$-power, $p^n$, in the base field, where $p$ is the characteristic. It is the opposite of a separable extension: the only elements that are both separable and purely inseparable over $K$ are those of $K$. Every algebraic extension splits into a separable part followed by a purely inseparable part, so the two kinds together describe all algebraic extensions. In characteristic $0$ the only purely inseparable extension of $K$ is $K$ itself.