Let F be a field of characteristic p and suppose that F L is separable and that p divides [L :

Let F be a field of characteristic p and suppose that F ⊂ L is separable and that p divides [L : F].

Suppose furthermore that any q-th root of unity, where q is prime and q ≡ 1 (mod p), that lies in L already lies in F.

Show that F ⊂ L cannot be solvable.