= Solution
For a <number field> $K$ and a <modulus of a number field> $\mathfrak m=\mathfrak m_0\mathfrak m_\infty$, let $I_K(\mathfrak m)$ be the group of <fractional ideals> prime to $\mathfrak m_0$, and let $P_{K,1}(\mathfrak m)$ consist of <principal fractional ideals> $(a)$ with $a\equiv1$ at the finite modulus and $a>0$ at the real places in $\mathfrak m_\infty$. The <ray class group> is $\operatorname{Cl}_{\mathfrak m}(K)=I_K(\mathfrak m)/P_{K,1}(\mathfrak m)$.
The <Main theorem of class field theory> states that there is a unique finite abelian <ray class field> $K_{\mathfrak m}/K$ for which the <Artin reciprocity map> induces
$$
\boxed{\operatorname{Cl}_{\mathfrak m}(K)\simeq
\operatorname{Gal}(K_{\mathfrak m}/K).}
$$
It is <unramified> outside the finite modulus and has no ramification at real places omitted from $\mathfrak m_\infty$. More generally, subgroups $H$ with $P_{K,1}(\mathfrak m)\subseteq H\subseteq I_K(\mathfrak m)$ correspond, with inclusions reversed, to intermediate abelian fields, with <Galois group> $I_K(\mathfrak m)/H$. Every finite abelian extension occurs for some modulus. Equivalently, $K_{\mathfrak m}$ is the maximal abelian extension with <conductor of an abelian extension> dividing $\mathfrak m$. This states both reciprocity and the <existence theorem of global class field theory>.
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