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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)


Class groups of quadratic fields

Author: Duncan A. Buell
Journal: Math. Comp. 30 (1976), 610-623
MSC: Primary 12A50; Secondary 12A25
MathSciNet review: 0404205
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Abstract: The author has computed the class groups of all complex quadratic number fields $ Q(\sqrt { - D} )$ of 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 = {\text{prime}}$, class group $ C(3) \times C(3) \times C(9.7)\quad (C(n)$ a cyclic group of order n), and $ D = 3640387 = 421.8647$, class group $ C(3) \times C(3) \times C(9.2)$. The author has also found polynomials representing discriminants of 3-rank $ \geqslant 2$, and has found 3-rank 3 for $ D = 6562327 = 367.17881,8124503,10676983,193816927$, all prime, $ 390240895 = 5.11.7095289$, and $ 503450951 = {\text{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 = {\text{prime}}$, class group $ C(15) \times C(15)$, had been found by A. O. L. Atkin; the next such D is $ 2075343 = 3.17.40693$, class group $ C(30) \times C(30)$. Finally, $ D = 3561799 = {\text{prime}}$ has class group $ C(21) \times C(63)$, the smallest D noncyclic for 3 and 7 together.

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PII: S 0025-5718(1976)0404205-X
Article copyright: © Copyright 1976 American Mathematical Society