= Idelic congruence subgroup
{title2=$C_K(\mathfrak m)$}
For a <modulus of a number field> $\mathfrak m$, form a subgroup $U(\mathfrak m)$ of the <idele group> as follows. At a finite prime dividing $\mathfrak m_0$ with exponent $n$, use $1+\mathfrak p^n\mathcal O_{K_{\mathfrak p}}$; at every other finite prime, use $\mathcal O_{K_{\mathfrak p}}^\times$. At a real place selected by $\mathfrak m_\infty$, use $\mathbb R_{>0}$, and at every other infinite place use the entire multiplicative completion.
The image of $U(\mathfrak m)$ in the <idèle class group> $C_K$ is the idelic congruence subgroup $C_K(\mathfrak m)$. Its quotient is the <ray class group>. The <unit> requirement outside the modulus is essential: allowing arbitrary <valuations> at those primes would discard <ideal>-class information. Multiplication by a <principal idele> accounts for changing a global generator while preserving its <ideal> class.
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