Ray class field

Here "Strahl" is German for ray, and often means the positive real line, which appears in the positivity conditions defining ray class groups.

Takagi proved the existence of the corresponding ray class fields in about 1920.

Chevalley reformulated the definition of ray class groups in terms of ideles in 1933.

The proof of existence of a ray class field of a given ray class group is long and indirect and there is in general no known easy way to construct it (though explicit constructions are known in some special cases such as imaginary quadratic fields).

in the idele class group of K. If K is the field of rational numbers, m is a nonzero rational integer, and S comprises the Archimedean place of K, then the ray class group of (m) and S is isomorphic to the group of units of Z/mZ, and the ray class field is the field generated by the mth roots of unity.