On the class numbers of real cyclotomic fields of conductor $pq$

TL;DR

This paper develops an extended computational method to determine class numbers of real cyclotomic fields with conductors that are products of two odd primes, overcoming limitations of previous approaches.

Contribution

It generalizes a method for prime conductors to composite conductors of two odd primes, enabling the calculation of class numbers for a broader class of fields.

Findings

Successfully computed class numbers for fields with conductor < 2000.

Determined the full order of the l-part of h+ for all odd primes l < 10000.

Extended applicability of class number computation methods to non-cyclic extensions.

Abstract

The class numbers $h^{+}$ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows and methods employing Leopoldt's decomposition of the class number become hard to use when the field extension is not cyclic of prime power. This is why other methods have been developed which approach the problem from different angles. In this paper we extend one of these methods that was designed for real cyclotomic fields of prime conductor, and we make it applicable to real cyclotomic fields of conductor equal to the product of two distinct odd primes. The main advantage of this method is that it does not exclude the primes dividing the order of the Galois group, in contrast to other methods. We applied our algorithm to real cyclotomic fields of conductor $<$ 2000 and we calculated the full order of the $l$-part of $h^{+}$ for all odd primes $l$ $<$ 10000.