Separable extension

Definition. An algebraic extension $L/K$ is separable if every element of $L$ is a separable element over $K$.

Equivalently (for finite extensions), $L/K$ is separable iff it can be generated by adjoining separable elements.

See also. perfect field , Galois extension , inseparable extension .

Examples.

1. Every algebraic extension of $\mathbb{Q}$ is separable (characteristic $0$).
2. Every finite field extension $\mathbb{F}_{p^n}/\mathbb{F}_p$ is separable.
3. The extension $\mathbb{F}_p(t^{1/p})/\mathbb{F}_p(t)$ is not separable (it is purely inseparable).