Let $K \leq M \leq E$ be field extensions, with $K \leq E$ separable.
Show that the extensions $K \leq M$ and $M \leq E$ are separable.
The extension $K\leq E$ is separable if all the elements in $E$ are separable. Correct?
Any help to continue would be appreciated.
Let $K,M,E$ be fields s.t. $K \subseteq M \subseteq E$. Then if $E/K$ is separable also $E/M$ is. This follows from the fact that the minimal polynomial of $\alpha \in E$ over $M$ divides the minimal polynomial of $\alpha$ over K.
The separability of extension the $M/K$ is even easier to see. (I think you can see this yourself just by looking at the definition)