Solution
= Solution
The ideal-theoretic <existence theorem of global class field theory> gives an inclusion-reversing correspondence between finite abelian extensions $L/K$ and congruence subgroups
$$
P_K(\mathfrak m)\subseteq H\subseteq I_K(\mathfrak m).
$$
The corresponding field satisfies
$$
H=\ker\!\left(I_K(\mathfrak m)
\xrightarrow{\operatorname{Art}_{L/K}}
\operatorname{Gal}(L/K)\right),
\qquad
I_K(\mathfrak m)/H\cong\operatorname{Gal}(L/K).
$$