Perfect fields and separability of finite extensions

A perfect field is a field $K$ such that every algebraic extension of $K$ is separable. The key consequence for field extensions is:

Theorem. If $K$ is perfect and $L/K$ is algebraic (in particular, if $L/K$ is finite), then $L/K$ is separable. Moreover, any algebraic extension $L$ of a perfect field $K$ is itself perfect.

Remarks

This is especially useful combined with the separable + normal = Galois criterion: over a perfect base field, to check that a finite extension is Galois, it suffices to check normality.

Examples

Characteristic $0$ . Every field of characteristic $0$ is perfect, so any finite extension of $\mathbb{Q}$ (e.g. $\mathbb{Q}(\sqrt2,\sqrt3)/\mathbb{Q}$ ) is automatically separable.

Finite fields. Any finite field $\mathbb{F}_{p^n}$ is perfect (see finite fields are perfect), hence any finite extension $\mathbb{F}_{p^m}/\mathbb{F}_{p^n}$ is separable.

A non-perfect base gives inseparability. Let $K=\mathbb{F}_p(t)$ . Then $K$ is not perfect, and $L=K(t^{1/p})$ is a finite (degree $p$ ) extension that is inseparable (its defining polynomial $x^p-t$ has zero derivative; see separable iff distinct roots).