Exercise 8, § 16

Let L be an extension of K. Suppose that K has characteristic $p >0$ and $L = K^{1/p}$. Prove that the group Br(L$/K$) is equal to the kernel of multiplication by $p$ in Br(K) (observe that if N is a finite Galois extension of K, then $N^{1/p}$ is a Galois extension of L with the same Galois group).