Efficient Computation of Class Numbers of Real Abelian Number Fields

摘要

Let {K_m} 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 P_m(x) = x~3 - mx~2 -(m + 3)x - 1. We develop two methods for computing the class numbers of these K_m's. As a byproduct of our computation, we found 32 cyclotomic fields Q(ζ_p) of prime conductors p < 10~(10) for which some prime q ≥ p divides the class numbers h_p~+ of their maximal real subfields Q(ζ_p)~+ (but we did not find any conterexample to Vandiver's conjecture!).