Proposition 9

For every Galois extension of finite degree L of K, the homomorphism $k_L$ is bijective.

Let $0 \in L^*$ be such that $0^n \in K$ and the residue class of $0^n \mod K^{*n}$ belongs to the kernel of $k_L$. For every $\sigma \in \mathrm{Gal}(L/K)$ we have by definition $\sigma(\theta) = 0$; hence $0 \in K^*$ and $\theta^n \in K^{*n}$. This proves the injectivity of $k_L$. Now let $f : \mathrm{Gal}(L/K) \to \mu_n(K)$ be a homomorphism; for all $\sigma, \tau \in \mathrm{Gal}(L/K)$ we have

By V, p. 65, Cor. 1 there exists $0 \in L^*$ such that $f(\sigma) = \sigma(\theta)/\theta$ for all $\sigma \in \mathrm{Gal}(L/K)$; since $f(\sigma)^n = 1$, we have $\sigma(\theta^n) = 0^n$ for all $\sigma \in \mathrm{Gal}(L/K)$, hence $0^n \in K^*$; if $a$ is the residue class of $\theta^n \mod K^{*n}$, we have by definition $f(\sigma) = \langle \sigma, a \rangle$ for $\sigma \in \mathrm{Gal}(L/K)$, hence $f = k_L(a)$.