# Inclusion between norm groups in the idèle class group

Reading Cassels and Fröhlich Chap. VII about Global Class Field Theory, I stumbled upon the following problem: if $$K\subset L\subset M$$ are finite abelian extensions, then the Main Theorem on Abelian Extensions (5.1, D) asserts the existence of the following commutative diagram $$\require{AMScd}$$ $$\begin{CD} C_K/N_{M/K}C_M @>{\psi_{M/K}}>> \text{Gal}(M/K)\\ @VVV @VVV\\ C_K/N_{L/K}C_L @>{\psi_{L/K}}>> \text{Gal}(L/K) \end{CD}$$ (where $$C_K$$ is the idèle class group of $$K$$, $$N_{M/K}$$ denotes the idèle norm and $$\psi_{M/K}$$ is the Artin map) and the author says the left vertical arrow comes from the inclusion $$N_{M/K}C_M\subset N_{L/K}C_L$$, but why does that inclusion hold? I think that from the fact that $$A_L\otimes_L M=A_M$$ (adèles) we have an inclusion of the idèles group $$J_L\subset J_M$$ but I cannot see why the norm seems to reverse that inclusion.

• This follows from the transitivity of norm on the idele class group: $$N_{M/K}(C_M) = N_{L/K}(N_{M/L}C_M)$$ Oct 24, 2018 at 9:39

As pointed out in the comments, we have $$N_{M/K} = N_{L/K} \circ N_{M/L},$$ so that $$N_{M/K}(C_M) = N_{L/K}(N_{M/L}(C_M)) \subset N_{L/K}(C_L).$$ This yields a quotient morphism $$C_K/N_{M/K}(C_M) \to C_K/N_{L/K}(C_L).$$