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

### Volume 164 / 2014

Acta Arithmetica 164 (2014), 381-397
MSC: Primary 11R29, 11R80; Secondary 11R18, 11Y40.
DOI: 10.4064/aa164-4-4

#### 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.