purely inseparable

Let $F$ be a field of characteristic $p>0$ and let $\alpha$ be an element which is algebraic over $F$. Then $\alpha$ is said to be purely inseparable over $F$ if ${\alpha }^{p^n}\in F$ for some $n\ge 0$.

An algebraic field extension $K/F$ is purely inseparable if each element of $K$ is purely inseparable over $F$.

Purely inseparable extensions have the following property: if $K/F$ is purely inseparable, and $A$ is an algebraic closure of $F$ which contains $K$, then any homomorphism $K\to A$ which fixes $F$ necessarily fixes $K$.

Let $K/F$ be an arbitrary algebraic extension. Then there is an intermediate field $E$ such that $K/E$ is purely inseparable, and $E/F$ is separable.

Example.

Let $s$ be an indeterminate, and let $K={𝔽}_3(s)$ where ${𝔽}_3$ is the finite field with $3$ elements. Let $F={𝔽}_3(s^6)$. Then $K/F$ is neither separable, nor purely inseparable. Let $E={𝔽}_3(s^3)$. Then $E/F$ is separable, and $K/E$ is purely inseparable.