Lemma 6

— Let L be a Galois extension of K. There exists a unique mapping $(\sigma, a) \mapsto \langle \sigma, a \rangle$ of $\mathrm{Gal}(L/K) \times ((L^n \cap K^*)/K^{*n})$ into $\mu_n(K)$ such that for every $\sigma \in \mathrm{Gal}(L/K)$ and every element $0 \in L^*$ such that $\theta^n \in K$ we have, on denoting by $\overline{\theta^n}$ the residue class of $\theta^n$ mod $K^{*n}$:

This mapping is bimultiplicative.

In effect the right-hand side of (13) is an n-th root of unity which only depends on the residue class mod $K^{*n}$ of $\theta^n$; this proves the first assertion. The second assertion is verified without difficulty.

For every Galois extension L of K let us write

for the homomorphisms derived from the above bimultiplicative mapping (V, p. 87).