Class groups of quadratic fields

Author:
Duncan A. Buell

Journal:
Math. Comp. **30** (1976), 610-623

MSC:
Primary 12A50; Secondary 12A25

DOI:
https://doi.org/10.1090/S0025-5718-1976-0404205-X

MathSciNet review:
0404205

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Abstract: The author has computed the class groups of all complex quadratic number fields of discriminant for . In so doing, it was found that the first occurrences of rank three in the 3-Sylow subgroup are , class group a cyclic group of order *n*), and , class group . The author has also found polynomials representing discriminants of 3-rank , and has found 3-rank 3 for , all prime, , and . 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. , class group , had been found by A. O. L. Atkin; the next such *D* is , class group . Finally, has class group , the smallest *D* noncyclic for 3 and 7 together.

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DOI:
https://doi.org/10.1090/S0025-5718-1976-0404205-X

Article copyright:
© Copyright 1976
American Mathematical Society