= Residue and signature group of a modulus
{title2=$G_{\mathfrak c}=(\mathcal O_K/\mathfrak c_0)^\times\times\{\pm1\}^{\mathfrak c_\infty}$}
This group records finite unit residues and signs at the selected real places of a <modulus of a number field>. Omit the residue factor for trivial finite modulus. Its quotient by the image of the <unit group> is naturally the <kernel> of the map from the <ray class group> to the ordinary <ideal class group>. The <weak approximation for number fields> theorem supplies elements realizing the specified data, and units account for changing a generator of a <principal fractional ideal>.
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