Show that every intermediate field is separable

Question

Let $K/k$ be a finite separable extension. I need to show that for every intermediate field $k ⊂ E ⊂ K$, the extensions $E/k$ and $K/E$ are separable

How can I try to achieve this?

Answer 1

Hint:

Based on the criterion that an extension field is separable if every element is a simple root of its minimal polynomial, that $E/k$ be separable if $/k$ is obvious, ince the minimal polynomial of the elements of $E$ is the same as their minimal polynomial as elements of $K$.

As to $K/EK$, just observe that the minimal polynomial of an element of $K$ over the field $E$ is a divisor (in $E[X]$) of its minimal polynomial over the field $k$.