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.

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

1. 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.

2. 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.

3. 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 ).