If $u\in F$ is separable over $K$, then $u$ is separable over $E$.

Question

I am studying studying separability and trying to solve this exercise problem from Hungerford.

Let $E$ be an intermediate field of the extension $K\subseteq F$. If $u\in F$ is separable over $K$, then $u$ is separable over $E$. Furthermore if $F$ is separable over $K$, then $F$ is separable over $E$ and $E$ is separable over $K$.

Here is the definition of separable:

Let $K$ be a field and $f\in K[x]$ an irreducible polynomial. The polynomial $f$ is said to be separable if in some splitting field of $f$ over $K$ every root of $f$ is a simple root.

Can anyone help me solving above exercise?

Answer 1

Hint:

The minimal polynomial of $u$ over $E$ is a divisor of its minimal polynomial over $K$.

This question will take care of the separability of $F$ over $R$. As to the separability of $E$ over $K$, the minimal polynomial of an element of $E$ over $K$ is the same as its minimal polynomial as an element of $F$.