= Ray class fields of the rational numbers
{title2=$\mathbb Q_{(m)}=\mathbb Q(\zeta_m+\zeta_m^{-1}),\quad\mathbb Q_{(m)\infty}=\mathbb Q(\zeta_m)$}
= Rational ray class fields
{synonym}
For a positive integer $m$, the finite <ray class group> of $\mathbb Q$ is $(\mathbb Z/m\mathbb Z)^\times/\langle-1\rangle$: 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 $\mathbb Q(\sqrt5)$, while $(5)\infty$ gives $\mathbb Q(\zeta_5)$. The <Artin reciprocity map> sends a prime $\ell\nmid m$ to the exponent $\ell$, taken modulo $\{\pm1\}$ in the finite case. Ramification at real places is possible only when those places are included in the <modulus of a number field>.
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