Class Numbers of Real Quadratic Number Fields by Ezra Brown

This article is a study of congruence conditions, modulo powers of two, on class number of real quadratic number fields Q(vu), for which d has at most thtee distinct prime divisors. Techniques used are those associated with Gaussian composition of binary quadratic forms. 1. Let hid) denote the class number of the quadratic field Qi\ß) and let h id) denote the number of classes of primitive binary quadratic forms of discriminant d [if d < 0 we count only positive forms]. It is well known [4] that hid) = h\d), unless d> 0 and the fundamental unit e of Qi\fd) has norm 1, in which case bid) =x/¡.h id). Recently many authors have studied conditions on d under which a given power of two divides hid) (see [3, References]). Most of these articles deal with imaginary fields; in this article, we shall treat real fields for which d has at most three distinct prime divisors. Our method used to study this problem is the method of composition of forms, used in [ll, [3]; we have included several known cases for the sake of completeness. 2. Preliminaries. A binary quadratic form is called ambiguous if its square, under Gaussian composition, is in the principal class, i.e. the class representing 1 (see [1] for explanations of any unfamiliar terminology). A class of forms is called ambiguous if it contains an ambiguous form. A form [a, b, c] = ax2 + bxy + cy is called ancipital if b = 0 or b = a. It was known to Gauss that the number of ambiguous classes of discriminant d equals the number of genera of discriminant d (see [7]), and that each ambiguous class of positive nonsquare discriminant contains exactly two ancipital forms with positive first coefficient (see [7]). The primitive forms of discriminant d form an abelian group G of order h id), the operation being composition and the identity being the principal class /. The principal genus G is a subgroup of G consisting of all the classes which are squares under composition; the index of G in G equals the number of genera. If d is the discriminant of a quadratic field, then d is fundamental, i.e. no square s2 > 1 exists for which d/s2 = 0 or 1 (mod 4). Hence the number of genera Received by the editors October 3, 1972. AMS (MOS) subject classifications (1970). Primary 10A15, 10C05, 12A25, 12A50.