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.

For $K=\mathbb{Q}(\sqrt{-6})$, reduction of $\mathbb{Z}[\sqrt{-6}]$ modulo $(\sqrt{-6})$ gives $\mathbb{Z}/6\mathbb{Z}$. Its unit group is covered by the units $\{1,-1\}$ of $K$. The ray class exact sequence therefore identifies the ray class group with the ordinary ideal class group, of order two. Consequently this ray class field is the Hilbert class field of Q of square root minus six. To verify its lack of ramification, the integral basis candidate $1,\sqrt{2},b,\sqrt{2}b$, where $b=(1+\sqrt{-3})/2$, has discriminant $8^2(-3{)}^2=576$. The relative discriminant formula also forces the field discriminant to be divisible by $(-24{)}^2=576$. Equality proves that the relative discriminant ideal is trivial.

The ring of integers is $\mathbb{Z}[\omega ]$, where ${\omega }^2+\omega +1=0$, and its class number is one by the Minkowski bound for ideal classes. Modulo seven it is ${\mathbb{F}}_7\times {\mathbb{F}}_7$. The six roots of unity reduce to the kernel of the product map on the two residue unit groups, so the ray class group is ${\mathbb{F}}_7^{\times }$, of order six. The relative Artin symbol on $K({\zeta }_7)$ is exponentiation by the norm of a fractional ideal modulo seven. It kills the principal ray subgroup, and $[K({\zeta }_7):K]=6$, proving that this is the ray class field.

For a positive integer $m$, the finite ray class group of $\mathbb{Q}$ is $(\mathbb{Z}/m\mathbb{Z}{)}^{\times }/⟨-1⟩$: the unit group permits either sign of a rational generator. Its ray class field is the maximal real subfield of the cyclotomic field $\mathbb{Q}({\zeta }_m)$. Including the real place requires positive principal generators and gives the full group $(\mathbb{Z}/m\mathbb{Z}{)}^{\times }$ and full cyclotomic field. Thus modulus $(5)$ gives $\mathbb{Q}(\sqrt{5})$, while $(5)\infty$ gives $\mathbb{Q}({\zeta }_5)$. The Artin reciprocity map sends a prime $\ell \nmid m$ to the exponent $\ell$, taken modulo $\{\pm 1\}$ in the finite case. Ramification at real places is possible only when those places are included in the modulus of a number field.