Definition:Purely Inseparable Field Extension

Definition
Let $E/F$ be an algebraic field extension.

Definition 1
The extension $E/F$ is purely inseparable every element $\alpha \in E \setminus F$ is inseparable.

Definition 2
Let $F$ have positive characteristic $p$.

The extension $E/F$ is purely inseparable for each $\alpha \in E$ there exists $n \in \N$ such that $\alpha^{p^n} \in F$.

Definition 3
Let $F$ have positive characteristic $p$.

The extension $E/F$ is purely inseparable each element of $E$ has a minimal polynomial of the form $X^{p^n} - a$.

Also see

 * Equivalence of Definitions of Purely Inseparable Extension
 * Definition:Relative Purely Inseparable Closure
 * Definition:Purely Inseparable Closure
 * Definition:Separable Field Extension
 * Definition:Inseparable Field Extension