Inseparable extension - Imported Knowlpedia Content

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

Definition (inseparable extension). The extension $K/F$ is inseparable if it is not a separable extension; equivalently, there exists $\alpha\in K$ whose minimal polynomial over $F$ is not separable (has a repeated root in a splitting field). This can only occur when the characteristic of $F$ is $p>0$ .

A particularly important special case is:

Definition (purely inseparable). Assume $\mathrm{char}(F)=p>0$ . The extension $K/F$ is purely inseparable if for every $\alpha\in K$ there exists $n\ge 0$ such that

\alpha^{p^n}\in F.

Equivalent characterizations

Equivalently, every $\alpha\in K$ is a root of a polynomial of the form $x^{p^n}-a$ with $a\in F$ , whose derivative is $0$ , so no such element is separable unless $\alpha\in F$ .

Examples

$K=\mathbb{F}_p(t^{1/p})$ over $F=\mathbb{F}_p(t)$ is purely inseparable since $(t^{1/p})^p=t\in F$ . The minimal polynomial $x^p-t$ has a repeated root.
More generally, $\mathbb{F}_p(t^{1/p^n})/\mathbb{F}_p(t)$ is purely inseparable of degree $p^n$ , with $(t^{1/p^n})^{p^n}=t$ .
If $F$ is an imperfect field of characteristic $p$ (i.e. not perfect), then choosing $a\in F\setminus F^p$ produces an inseparable extension $F(a^{1/p})/F$ .