Exercise 18, § 1

a) Let L be a (commutative) field. Let G be a group of automorphisms of L, and K the field of invariants of G. Then every element $g$ of G acts on $W(L)$ (by $W(g)$) and, for every integer $n \geqslant 1$, on $W_n(L)$. The set of elements of $W(L)$ (resp. $W_n(L)$) invariant under the action of G is $W(K)$ (resp. $W_n(K)$).

b) Suppose that L is a finite galois extension of K, and let us denote by $\mathbf{Z}[G]$ the algebra over $\mathbf{Z}$ of the (finite) group G. For every $\mathbf{Z}$-module M, let us denote by $M^G$ the $\mathbf{Z}[G]$-module $\mathrm{Hom}_\mathbf{Z}(\mathbf{Z}[G], M)$ (endowed with its natural structure of left $\mathbf{Z}[G]$-module). Prove that the $\mathbf{Z}[G]$-module $W(L)$ (resp. $W_n(L)$) is isomorphic to $W(K)^G$ (resp. $W_n(K)^G$). Deduce that one has

for every integer $i > 0$. (Use the normal basis theorem A, V, p. 70, Theorem 6 and A, X, p. 111 to 113.)