Separable extension

Let $K/F$ be an algebraic field extension .

Definition (separable extension). The extension $K/F$ is separable if every element $\alpha\in K$ is a separable element over $F$.

When $K/F$ is finite, separability admits useful equivalent formulations. For example, if $\overline F$ is an algebraic closure of $F$, then $K/F$ is separable iff there are exactly $[K:F]$ distinct $F$-embeddings $K\hookrightarrow \overline F$. Separability behaves well in towers, as summarized in separability in towers .

Examples.

1. Every algebraic extension of a characteristic $0$ field is separable. In particular, $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is separable (even though it is not normal ).
2. For finite fields, $\mathbb{F}_{p^n}/\mathbb{F}_p$ is separable; in fact it is Galois .
3. The extension $\mathbb{F}_p(t^{1/p})/\mathbb{F}_p(t)$ is not separable: the element $t^{1/p}$ is not separable over $\mathbb{F}_p(t)$.