Real cyclotomic fields of prime conductor and their class numbers

TL;DR

This paper advances the understanding of class numbers in cyclotomic fields by establishing new results for prime conductors between 71 and 241, using novel techniques that extend to other totally real fields.

Contribution

It proves the plus part of the class number is 1 for prime conductors 71 to 151 and, under GRH, determines class numbers for 167 to 241, generalizing to other totally real fields.

Findings

Plus part of class number is 1 for primes 71-151

Class numbers determined under GRH for primes 167-241

Technique applicable to other totally real fields

Abstract

Surprisingly, the class numbers of cyclotomic fields have only been determined for fields of small conductor, e.g. for prime conductors up to 67, due to the problem of finding the "plus part," i.e. the class number of the maximal real subfield. Our results have improved the situation. We prove that the plus part of the class number is 1 for prime conductors between 71 and 151. Also, under the assumption of the generalized Riemann hypothesis, we determine the class number for prime conductors between 167 and 241. This technique generalizes to any totally real field of moderately large discriminant, allowing us to confront a large class of number fields not previously treatable.