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. 

yes, consider any embedding f of L into algebraic closure of K.f can be extended to embedding of K into algebraic closure.as K is normal that embedding maps K into itself.
   let,x be any element of L then f(x) is in K.as x is separable implies f(x) is also separable so f(x) belongs to L so any embedding of L into algebraic closure is automorphism of L.So L is normal.
