Definition:Purely Inseparable Field Extension
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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 $X^{p^n} - a$.