Let $I_K(\mathfrak m)$ be the group of fractional ideals prime to $\mathfrak m_0$. Let $P_K(\mathfrak m)$ consist of principal ideals $(\alpha)$ with
$$
\alpha\equiv1\pmod{\mathfrak p^{n_\mathfrak p}}
$$
for every $\mathfrak p\mid\mathfrak m_0$ and $\sigma(\alpha)>0$ for every real place $\sigma\mid\mathfrak m_\infty$. The <ray class group> is
$$
\boxed{\operatorname{Cl}_{\mathfrak m}(K)
=I_K(\mathfrak m)/P_K(\mathfrak m).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 1 / b / Solution

{title2=$\operatorname{Cl}_{\mathfrak m}(K)$}
{wiki}

The ray class group modulo $\mathfrak m$ is $I_K(\mathfrak m)/P_K(\mathfrak m)$, where the numerator consists of fractional ideals prime to $\mathfrak m_0$ and the denominator consists of $(\alpha)$ with $\alpha\equiv1$ at the finite modulus and positive at every real place in $\mathfrak m_\infty$.


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 Incoming links: Ray class group