ray class groups in $\Bbb{Q}$ I study class field theory from the book "Primes of the form $x^2+ny^2$", D. Cox. I want to find ray class groups in $\Bbb{Q}$. 
Let $K$ be a number field, $\mathfrak{m}$ be a modulus of $K$. In the book, $I_K(\mathfrak{m})$ is defined as the group of all fractional ideals of $K$ coprime to $\mathfrak{m}_0$ (finite part of $\mathfrak{m}$). $P_K(\mathfrak{m})$ is defined as the subgroup of $I_K(\mathfrak{m})$ generated by the principle ideals $\alpha\mathcal{O}_K$, where $\alpha\in\mathcal{O}_K$ satisfies $\alpha\equiv 1 \ mod \ \mathfrak{m}_0$ and $\sigma(\alpha)> 0$ for every real infinite prime $\sigma$ dividing $\mathfrak{m}_0$ (infinite part of $\mathfrak{m}$).
For example, Set $K=\Bbb{Q}$ and $\mathfrak{m}=(8)$. Then according to the above definitions, I find $$I_\Bbb{Q}((8))=\{(a/b)\Bbb{Z}: gcd(a,8)=gcd(b,8)=1 \}=\{(a/b)\Bbb{Z}: 2\nmid a,b\}$$
$$P_\Bbb{Q}((8))=<a\Bbb{Z}: a\in\Bbb{Z},\  a\equiv 1\ mod\ (8)>=\{(a/b)\Bbb{Z}: a\equiv 1\ mod\ 8, \ b\equiv 1\ mod\ 8\}$$
First , I am not sure that I write these groups correctly. Also, I couldn`t conclude that the ray class group
$Cl_\Bbb{Q}((8)):=I_\Bbb{Q}((8))/P_\Bbb{Q}((8))$ is isomorphic to $(\Bbb{Z}/4\Bbb{Z})^*$. Generally, I have trouble to describe the ray class groups. 
Thank you

The ray class group modulo $8$ corresponds to the ray class field modulo $8$, which is the maximal totally real abeliab extension of ${\mathbb Q}$ with conductor $8$, i.e., the maximal real subfield of ${\mathbb Q}(\zeta_8)$, namely ${\mathbb Q}(\sqrt{2})$. This is a quadratic extension, as is confirmed by the fact that the ray class group modulo $8$ has order $2$. In fact, the ideal generated by $(3)$ in the ray class group is nor principal, i.e., is not generated by an element $a/b \equiv 1 \bmod 8$. Moreover $(3) = (5)$ since $(5) = (-5) = (3)$ inside the ray class group. Ray class groups over the rationals are basically residue class groups since ${\mathbb Q}$ has class number $1$.
