Efficient computation of class numbers of real abelian number fields

Abstract

Let {Km} be a parametrized family of real abelian number fields of known regulators, e.g. the simplest cubic fields associated with the Q-irreducible cubic polynomials Pm(x) = x3 − mx2 − (m + 3)x − 1. We develop two methods for computing the class numbers of these Km’s. As a byproduct of our computation, we found 32 cyclotomic fields Q(ζp) of prime conductors p < 1010 for which some prime q ≥ p divides the class numbers h+ p of their maximalrealsubfiel ds Q(ζp)+ (but we did not find any conterexample to Vandiver’s conjecture!).