Class numbers of totally real fields and applications to the Weber class number problem

TL;DR

This paper introduces a new technique for determining the class number of large discriminant totally real fields, enabling progress on Weber's class number problem concerning real cyclotomic fields.

Contribution

A novel method is developed to compute class numbers of difficult totally real fields, advancing the study of their arithmetic properties and addressing longstanding conjectures.

Findings

Successfully applied the method to specific fields

Provided evidence supporting Weber's class number conjecture

Enhanced understanding of class number behavior in large discriminant fields

Abstract

The determination of the class number of totally real fields of large discriminant is known to be a difficult problem. The Minkowski bound is too large to be useful, and the root discriminant of the field can be too large to be treated by Odlyzko's discriminant bounds. We describe a new technique for determining the class number of such fields, allowing us to attack the class number problem for a large class of number fields not treatable by previously known methods. We give an application to Weber's class number problem, which is the conjecture that all real cyclotomic fields of power of 2 conductor have class number 1.