Separability in Towers

Theorem.
Let $K \subseteq L \subseteq M$ be a tower of fields with $M/K$ algebraic.

1. If $M/K$ is separable , then $M/L$ and $L/K$ are separable.
2. If $L/K$ and $M/L$ are separable, then $M/K$ is separable.

In the finite case, these statements combine naturally with the tower law for degrees.

Examples.

1. If $K$ has characteristic $0$, then every algebraic extension is separable; hence all steps in any algebraic tower over $K$ are separable.
2. Finite fields: $\mathbb{F}_p \subseteq \mathbb{F}_{p^d} \subseteq \mathbb{F}_{p^n}$ are all separable extensions (see finite fields are perfect ).
3. If $L/K$ is separable and $\alpha$ is separable over $L$, then $\alpha$ is separable over $K$, so $L(\alpha)/K$ is separable.

Related. perfect base ⇒ separable finite extensions , separable ⇔ distinct roots .