We contribute to the theory of orders of number fields. We define a notion of
ray classgroup associated to an arbitrary order in a number field together with an arbitrary ray
class modulus for that order (including Archimedean data), constructed using invertible
fractional ideals of the order. We show existence of
ray class fields corresponding to
the class groups. These ray class groups (resp., ray class fields) specialize to classical
ray class groups (resp., fields) of a number field in the case of the maximal order, and
they specialize to ring class groups (resp., fields) of orders in the case of trivial modulus.
We give exact sequences for simultaneous change of order and change of modulus. As a
consequence, we identify the ray class field of an order with a given modulus as a specific
subfield of a ray class field of the maximal order with a larger modulus. We also uniquely
describe each ray class field of an order in terms of the splitting behavior of primes.
Keywords
class field theory, orders of number fields, ray class
fields, ring class fields
