Idele class group

Summary

The idele class group of a number field $K$ is the quotient $C_K = I_K / K^\times$ of the idele group $I_K$ by the nonzero elements of $K$. The idele group $I_K$ is the group of units of the adele ring of $K$, with its own restricted product topology. In class field theory, the Artin reciprocity map goes from $C_K$ onto the Galois group of the maximal abelian extension of $K$. The finite abelian extensions of $K$ then match the open subgroups of finite index of $C_K$.