Separable closure and normality Let $K/F$ be a normal algebraic extension and let $L = (K/F)^{sep}$ be the subfield of elements of $K$ which are separable over $F$ (this is also called the separable closure of $F$ in $K$). Is $L/F$ necessarily normal?
I really do not have a clue about this question so any hints will be appreciated. 

Take $a\in L$ and let $p$ be its minimal polynomial. Since $a\in K$ and $K$ is normal, $p$ is a product of linear factors :
$$p=\prod_{j=1}^n(X-a_j),$$
(where $a=a_1$) and by definition of $L$ all the $a_j$'s are distinct. They have the same minimal polynomial $p$, so all of them are in $L$; hence $L$ is normal.
