Ray class fields of the rational numbers (${\mathbb{Q}}_{(m)}=\mathbb{Q}({\zeta }_m+{\zeta }_m^{-1}),{\mathbb{Q}}_{(m)\infty }=\mathbb{Q}({\zeta }_m)$)

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.