= Congruence subgroup of fractional ideals
{title2=$P_K(\mathfrak m)\subseteq H\subseteq I_K(\mathfrak m)$}
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.
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