Ray class exact sequence

For a finite modulus $\mathfrak{m}$, there is an exact sequenceWith real places in the modulus, the residue-unit term and the unit map also record the required signs.

For a modulus with no infinite component, suppose the ordinary ideal class group is trivial. Then the ray class group is trivial exactly when the global unit group maps surjectively onto the product of local unit residue groups at the finite modulus. Negative real units are allowed: positivity restrictions occur only when the modulus includes the corresponding real places.

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.