Inseparable extension

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

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.

1. $K=\mathbb{F}_p(t^{1/p})$ 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.
2. More generally, $\mathbb{F}_p(t^{1/p^n})/\mathbb{F}_p(t)$ is purely inseparable of degree $p^n$, with $(t^{1/p^n})^{p^n}=t$.
3. 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 $F(a^{1/p})/F$.