Idèle class group ($C_K$)

The idele class group is $C_K={\mathbb{I}}_K/K^{\times }$, where $K^{\times }$ is embedded diagonally.

The ideles congruent to one modulo $\mathfrak{m}$ have finite components in $1+{\mathfrak{p}}^{n_{\mathfrak{p}}}$ at primes in the finite modulus and positive components at selected real places. Their image in $C_K$ is the idelic congruence subgroup.

For a finite abelian extension $L/K$, the norm group is the image of the idele-class norm $C_L\to C_K$. Class field theory gives $C_K/N_{L/K}{C}_L\cong \mathrm{Gal}(L/K)$.

The $v$-component of the idele norm is ${\prod }_{w\mid v}{N}_{L_w/K_v}(x_w)$. It descends to an idèle class group norm $C_L\to C_K$.

For a finite Galois extension $L/K$ with group $G$, the kernel of the global reciprocity map $C_K\to G^{\mathrm{ab}}$ is $N_{L/K}{C}_L$. It follows that this norm group equals the norm group from the maximal abelian subextension $L^{[G,G]}/K$.