Hint for (a): Say L' is such that L'/F is Galois and L is contained in L'. Let L_1,...,L_n be all fields of the form h(L) where h is an automorphism in Gal(L',F). Prove that the composite L_1...L_n is the Galois closure of L and its degree is a power of p. Do this first for n=2 where you use the diamond diagram from Proposition 19 on p. 591. Then go by induction on n.

This is a more demanding problem, therefore is marked by *.