Corollary

If L is a Galois extension of K, the homomorphism $k_L$ is injective and its image is the group $\mathrm{Hom}_c(\mathrm{Gal}(L/K), \mu_n(K))$ of continuous homomorphisms of the topological group $\mathrm{Gal}(L/K)$ into the discrete group $\mu_n(K)$.

This follows at once from what has been said, using the fact that L is an increasing directed union of Galois extensions $L_i$ of finite degree and that the homomorphism of $\mathrm{Gal}(L/K)$ into $\mu_n(K)$ is continuous if and only if it can be factored through one of the quotients $\mathrm{Gal}(L_i/K)$ of $\mathrm{Gal}(L/K)$.