Inseparable extension

Definition. An algebraic extension $L/K$ is inseparable if it is not a separable extension , i.e. if some element of $L$ is not separable over K .

Inseparability can occur only in characteristic $p>0$ (see characteristic ). A common special case is a purely inseparable extension, where every element $\alpha\in L$ satisfies $\alpha^{p^n}\in K$ for some $n$.

See also. perfect field , Frobenius endomorphism .

Examples.

1. Let $K=\mathbb{F}_p(t)$ and $L=\mathbb{F}_p(t^{1/p})$. Then $L/K$ is inseparable: $t^{1/p}$ has minimal polynomial $x^p-t$ with repeated roots.
2. More generally, $\mathbb{F}_p(t^{1/p^n})/\mathbb{F}_p(t)$ is purely inseparable for any $n\ge 1$.
3. No nontrivial inseparable algebraic extension exists over $\mathbb{Q}$ (characteristic $0$).