Ray class field

In mathematics, a ray class field is an abelian extension of a global field associated with a ray class group of ideal classes or idele classes. Every finite abelian extension of a number field is contained in one of its ray class fields. The term "ray class group" is a translation of the German term "Strahlklassengruppe". Here "Strahl" is German for ray, and often means the positive real line, which appears in the positivity conditions defining ray class groups. uses "Strahl" to mean a certain group of ideals defined using positivity conditions, and uses "Strahlklasse" to mean a coset of this group. There are two slightly different notions of what a ray class field is, as authors differ in how the infinite primes are treated. Weber introduced ray class groups in 1897. 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. If m is an ideal of the ring of integers of a number field K and S is a subset of the real places, then the ray class group of m and S is the quotient group where Im is the group of fractional ideals co-prime to m, and the "ray" Pm is the group of principal ideals generated by elements a with a ≡ 1 mod m that are positive at the places of S. When S consists of all real places, so that a is restricted to be totally positive, the group is called the narrow ray class group of m. Some authors use the term "ray class group" to mean "narrow ray class group". A ray class field of K is the abelian extension of K associated to a ray class group by class field theory, and its Galois group is isomorphic to the corresponding ray class group. 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).