The Stacks project

Definition 9.14.1. Let $F$ be a field of characteristic $p > 0$. Let $K/F$ be an extension.

1. An element $\alpha \in K$ is purely inseparable over $F$ if there exists a power $q$ of $p$ such that $\alpha ^ q \in F$.

2. The extension $K/F$ is said to be purely inseparable if and only if every element of $K$ is purely inseparable over $F$.

If we have a field extension $L/M$ (with no condition on the characteristic of $M$), then we will say the extension is purely inseparable if either $L = M$ or the characteristic of $M$ is a prime number $p$ and $L/M$ is purely inseparable in the sense defined above.