Fitting Ideals of Class Groups of Real Fields with Prime Power Conductor

✍ Scribed by Pietro Cornacchia; Cornelius Greither

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 254 KB
• Volume: 73
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1998.2300

✦ Synopsis

For a totally real field of prime power conductor, we determine the Fitting ideal over the Galois group ring of the ideal class group and of the narrow ideal class group.

1998 Academic Press

1. Introduction

In this paper we prove a structure result on the ideal class group and on the narrow ideal class group of totally real fields with prime power conductor. For any number field K we denote its ideal class group by Cl K , its narrow ideal class group by Cl K , its unit group by O* K and the group of totally positive units by O* K, + . Let l be a prime number, let n # N and suppose that K has conductor l n , that is to say K is contained in the field Q(ln) obtained by adjoining to Q a primitive l n th root of unity ln and n is minimal. We denote by G the Galois group Gal(KÂQ). We define the group of cyclotomic units Cyc K of K as Cyc K =O* K & (1&ln) Z[Gal(Q(ln)ÂQ)] .

Article No. NT982300