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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.

The existence theorem of global class field theory, together with reciprocity, gives an inclusion-reversing bijection between finite abelian extensions $L/K$ inside a fixed algebraic closure and open finite-index subgroups of the idèle class group $C_K$. The subgroup for $L$ is $N_{L/K}{C}_L$, andEquivalently, for each modulus $\mathfrak{m}$, subgroups between the ray-principal group $P_K(\mathfrak{m})$ and the ideal group $I_K(\mathfrak{m})$ classify abelian extensions whose conductor divides $\mathfrak{m}$. Their quotients are identified with the corresponding Galois groups by the Artin map. Changing the modulus gives the same extension when the induced congruence subgroup is the same.

The conductor criterion says that a finite place has conductor exponent zero exactly when the extension is unramified there; a real place is absent from the conductor exactly when it stays real. Thus conductor one means unramified at every finite place and no real-to-complex ramification.

Apply the existence theorem to the open subgroup of $C_K$ which is the image ofThe idele-to-ideal map identifies the quotient by this subgroup with the ordinary ideal class group, which is finite. Therefore its class field $H$ exists and satisfiesIts conductor is one, so it is everywhere unramified. Conversely any everywhere-unramified abelian extension has conductor one; its Artin map kills $U$, hence factors through this quotient and corresponds to an extension contained in $H$. This proves that $H$ is exactly the Hilbert class field, the maximal abelian everywhere-unramified extension. Here the ordinary ideal class group is appropriate because the infinite factors include all $K_v^{\ast }$, rather than only the positive real factors used in a narrow ray class group.

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 $a\equiv 1$ at the finite modulus and $a>0$ at the real places in ${\mathfrak{m}}_{\infty }$. The ray class group is ${\mathrm{Cl}}_{\mathfrak{m}}(K)=I_K(\mathfrak{m})/P_{K,1}(\mathfrak{m})$.

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 inducesIt 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.

The ideal-theoretic existence theorem of global class field theory gives an inclusion-reversing correspondence between finite abelian extensions $L/K$ and congruence subgroupsThe corresponding field satisfies

The subgroup $H/P_{\mathbb{Q}}(8\infty )=\{1,7\}$ corresponds inside $\mathbb{Q}({\zeta }_8)$ to the fixed field of $\{1,c\}$, where $c$ is complex conjugation. This is the maximal real subfieldThus the existence theorem of global class field theory associates $H$ with $\boxed{\mathbb{Q}(\sqrt{2})}$.