nLab separable field extension

Definition

A polynomial $P$ over a field $K$ is separable if all its irreducible factors have distinct roots over the algebraic closure $\bar{K}$ of $K$.

An extension $K\subset L$ of fields is separable if every element $x\in L$ is a root of a separable polynomial over $K$.

Properties

Every finite separable field extension is an étale morphism of rings.

If $K\subset L\subset M$ are fields and $K\subset M$ is separable, then $L\subset M$ is also separable.