Proposition 11

*Let $L$ be a finite-degree galois extension of a commutative field $K$, with galois group $G$. a) One has $H^i(G, L) = 0$ for $i \geqslant 1$. b) One has $H^1(G, L^*) = 0$. c) The group $H^2(G, L^*)$ is canonically isomorphic to the group $\mathrm{Br}(K, L)$ (VIII, § 13).

The normal basis theorem (V, §10, No. 9, Theorem 6) shows that $L$ is isomorphic as a $\mathbf{Z}^{(G)}$-module to $K^G = \mathrm{Hom}_{\mathbf{Z}}(\mathbf{Z}^{(G)}, K)$; assertion a) then follows from Lemma 1. In view of Example 2, assertion b) follows from V, §10, No. 5, Corollary 1 to Proposition 9: finally, assertion c) was proved in VIII, §13.