Corollary

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

This is proved in the same way as the Cor. of Prop. 9, V, p. 89.