Congruence subgroup of fractional ideals ($P_K(\mathfrak{m})\subseteq H\subseteq I_K(\mathfrak{m})$)

A subgroup of the group of fractional ideals prime to a modulus of a number field, containing the principal ray subgroup. It is the inverse image of a subgroup of the ray class group. The existence theorem of global class field theory realizes it as the kernel of an Artin map, and identifies its quotient with the corresponding Galois group. This is the ideal-theoretic class-field meaning of congruence subgroup.