Computing ray class groups, conductors and discriminants

We use the algorithmic computation of exact sequences of Abelian groups to compute the complete structure of (ZK /m)∗ for an ideal m of a number field K, as well as ray class groups of number fields, and conductors and discriminants of the corresponding Abelian extensions. As an application we give several number fields with discriminants less than previously known ones. The paper is divided as follows. In §1, we give a complete algorithm for computing the groups (ZK /m)∗ for a number field K and an arbitrary modulus m. In §2, we describe the tools necessary for the determination of the ray class group of a number field, and also for solving the corresponding principal ideal problem. In §3, we explain how to compute signatures, conductors and discriminants of the fields associated to subgroups of the ray class group by global class field theory. In principle we can give relative and absolute discriminants of all Abelian extensions of a given base field. Finally in §4, w...

• Type: Journal
• Year: 1998
• Where: MOC
• Authors: Henri Cohen, Francisco Diaz y Diaz, Michel Olivier
• Keywords: Abelian Extensions, Corresponding Abelian Extensions, MOC 1998, Ray Class Group