Class Groups of Quadratic Fields

The author has computed the class groups of all complex quadratic number fields Q(\f^~D) °f discriminant D for 0 < D < 4000000. In so doing, it was found that the first occurrences of rank three in the 3-Sylow subgroup are D = 3321607 = prime, class group C(3) x C(3) x C(9.7) (C(n) a cyclic group of order n), and D = 3640387 = 421.8647, class group C(3) X C(3) X C(9.2). The author has also found polynomials representing discriminants of 3-rank > 2, and has found 3-rank 3 for D 6562327 = 367.17881, 8124503, 10676983, 193816927, all prime, 390240895 = 5.11.7095289, and 503450951 = prime. The first five of these were discovered by Diaz y Diaz, using a different method. The author believes, however, that his computation independently establishes the fact that 3321607 and 3640387 are the smallest D with 3-rank 3. The smallest examples of noncyclic 13-, 17-, and 19-Sylow subgroups have been found, and of groups noncyclic in two odd p-Sylow subgroups. D = 119191 = prime, class group C(15) X C(15), had been found by A. O. L. Atkin; the next such D is 2075343 = 3.17.40693, class group C(30) X C(30). Finally, D = 3561799 = prime has class group C(21) X C(63), the smallest D noncyclic for 3 and 7 together. Introduction. Throughout this paper, —D < 0 will denote the discriminant of an imaginary quadratic number field, and "smallest" will refer to D, not to D, so that "smallest D" means "largest discriminant." The author has computed the class groups of all quadratic number fields Q(y/-d), d > 0, of discriminant D, for 0 < D< 4000000. By a theorem of Gauss, if D has k distinct prime factors, the 2-Sylow subgroup of the class group has rank k — 1. Apart from this, the groups tend to be cyclic. Even the 2-Sylow subgroup tends to be k 2 elementary 2-groups and one large cyclic factor collecting the other powers of two in the class number, so that the 2-Sylow subgroup of the subgroup of squares is cyclic. In computing the 2-Sylow subgroup, then, we actually computed that subgroup of the subgroup of squares, and shall, by abuse of language, call this the 2-Sylow subgroup, calling the group cyclic if the subgroup of squares is. The subgroup of squares is, in the terminology of Gauss, the principal genus, and a discriminant for which the principal genus is noncyclic is called irregular. Thus, what we call a discriminant with a noncyclic 2-Sylow subgroup is a discriminant which Gauss would call irregular. Statistics were kept on the frequency of occurrence of noncyclic groups, and of the noncyclic p-Sylow subgroups for p = 2, 3, 5, 7. Special listings were also made of the noncyclic p-Sylow subgroups for p > 11, of the p-Sylow subgroups C(pa) x C(pb) with a and b>2, and of the class groups noncyclic in more than one p-Sylow subgroup. We note here that 95.74% of the class groups turned out to be cyclic. The Received June 3, 1975; revised October 14, 1975. AMS (MOS) subject classifications (1970). Primary 12-04, 12A25. Copyright © 1976. American Matricm.it,,al Society