Perfect Base Field ⇒ Finite Extensions are Separable

Theorem.
If $K$ is a perfect field and $L/K$ is a finite field extension , then $L/K$ is separable .

More generally, every algebraic extension of a perfect field is separable.

Examples.

1. If $\mathrm{char}(K)=0$, then $K$ is perfect, so every finite extension $L/K$ is separable.
2. If $K=\mathbb{F}_p$ (or any finite field), then $K$ is perfect (see finite fields are perfect ), hence $\mathbb{F}_{p^n}/\mathbb{F}_p$ is separable.
3. Over a perfect field $K$, the splitting field of any polynomial is automatically separable over $K$ as soon as the polynomial is separable (link: separable ⇔ distinct roots ).

Related. separability in towers .