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 $C_K/N_{L/K}{C}_L\cong \mathrm{Gal}(L/K)$.