Introduction

For the class number of an abelian numberfield K we have a decomposition h = h+.h−, where h+ is the class number of the maximal real subfield K+ of K, and h− is a positive integer, for which there exists an explicit formula. In this lecture we are concerned with the determination of h+, i.e. the determination of the class number of a real abelian number field.

For an abelian number field K the conductor f(K) is defined as the smallest positive integer f for which K ⊂ ℚ(ζf), with ζf a primitive f-th root of unity. In this lecture we show how to compute the class numbers of most real abelian number fields of conductor ≤ 200, in some cases assuming the generalized Riemann hypothesis. For more details see [3].

The results

In theorems 1, 2, 3 and 4 we list our results. By GRH we denote the generalized Riemann hypothesis for the zeta-function of the Hilbert class field of Q(ζf(k)). The Euler function is denoted byø.

Theorem 1 Suppose that f(K) = q is a prime power. Then h(K) = 1 if ø(q) ≤ 66.

Theorem 2 Suppose that f(K) = q is a prime power, and assume GRH. Then

h(K) = 4 if q = 163

h(K) = 1 for all other K for which ø(q) ≤ 162.

Suppose next that f(K) = n is not necessarily a prime power. The genus field G(K) of K is defined as the maximal totally unramified extension of K which is abelian over ℚ.