Class number computations of real abelian number fields

In this paper we describe the calculation of the class numbers of most real abelian number fields of conductor ? 200. The technique is due to J. M. Masley and makes use of discriminant bounds of A. M. Odlyzko. In several cases we have to assume the generalized Riemann hypothesis. Introduction. It is well known that the class number h of an abelian number field can be written as h = h h -, where h ? is the class number of the maximal real subfield K? of K and h is an integer. We can determine the relative class number h in a straightforward way, using the complex analytic class number formula (see [7, Kap. III], or [9, Chapter 3, Section 3]). For the full cyclotomic fields Q(n), with 4O(n) < 256, and their subfields, one can deduce hfrom the tables of G. Schrutka von Rechtenstamm [15]; here Dn denotes a primitive nth root of unity, and 40 is the Euler function. For the class number factor h + the complex analytic class number formula is less useful, since it requires that the units of K? be known. Alternative techniques have been developed by J. M. Masley [13], who computed the class number of almost all real cyclic number fields of conductor < 100; here the conductor of K is the least f for which K C Q(tf ). In this paper we apply Masley's techniques, with a few additions, to determine the class numbers of a large collection of real abelian fields of conductor < 200; see Section 1 for a precise statement of our results, some of which assume the generalized Riemann hypothesis. An important ingredient of Masley's method is the use of discriminant lower bounds proved by A. M. Odlyzko [14]. These lead to an upper bound for the class number of a real abelian number field, provided that its conductor, or more precisely its root discriminant (see [13, Section 1]), is sufficiently small. It follows that this method can only be used for a finite number of real abelian number fields. The existence of infinite class field towers shows that this remains true after any future improvement of Odlyzko's bounds. In fact, examples of J. Martinet [12] show that the method will never apply to fields whose root discriminant is larger than five times the present bound, under assumption of the generalized Riemann hypothesis. The structure of this paper is as follows. Section 1 contains our results and Section 2 lists the theorems used in the proofs. The proofs themselves are largely suppressed. Received October 24, 1980; revised December 24, 1981. 1980 Mathematics Subject Classification. Primary 12-04, 12A35, 12A55.