Exercise 10, § 16

a) Let L be a cyclic extension of K. Deduce from Exercises 9 and 7 that the groups Br(L$/K$) and $K^*/N_{L/K}(L^*)$ are isomorphic. b) Suppose that every extension L of K of finite degree is cyclic and that the norm mapping $N_{L/K}: L^*\rightarrow K^*$ is surjective. Prove that every field extension of K of finite degree with center K is equal to K.

c) Apply the previous question to finite fields.