Ideal class groups of cyclotomic number fields I

TL;DR

This paper generalizes divisibility properties of minus class numbers from cyclotomic fields to CM-fields using class field theory, providing more direct proofs and additional results on Hasse's unit index.

Contribution

It extends known results on class number divisibility to CM-fields and offers more direct proofs and new insights on Hasse's unit index.

Findings

Divisibility properties of minus class numbers generalized to CM-fields

New proofs for Hasse's unit index in CM-fields

Enhanced understanding of class field theory applications

Abstract

Following Hasse's example, various authors have been deriving divisibility properties of minus class numbers of cyclotomic fields by carefully examining the analytic class number formula. In this paper we will show how to generalize these results to CM-fields by using class field theory. Although we will only need some special cases, we have also decided to include a few results on Hasse's unit index for CM-fields as well, because it seems that our proofs are more direct than those given by Hasse.