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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.

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

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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 $$C_K/N_{M/K}(C_M) \to C_K/N_{L/K}(C_L).$$

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