Existence theorem of global class field theory

Finite abelian extensions of $K$ correspond contravariantly to congruence subgroups $H$ with $P_K(\mathfrak{m})\subseteq H\subseteq I_K(\mathfrak{m})$, and the Artin map identifies $I_K(\mathfrak{m})/H$ with the Galois group.

The ray class field modulo $\mathfrak{m}$ is the maximal abelian extension whose Artin map factors through the ray class group; its degree is the ray class number.