# Definition:Purely Inseparable Field Extension

## Definition

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

### Definition 1

The extension $E/F$ is purely inseparable if and only if every element $\alpha \in E \setminus F$ is inseparable.

### Definition 2

Let $F$ have positive characteristic $p$.

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