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Class groups of the quadratic fields found by F. Diaz y Diaz


Author: Daniel Shanks
Journal: Math. Comp. 30 (1976), 173-178
MSC: Primary 12A25; Secondary 12A50
DOI: https://doi.org/10.1090/S0025-5718-1976-0399039-9
Corrigendum: Math. Comp. 30 (1976), 900.
Corrigendum: Math. Comp. 30 (1976), 900.
MathSciNet review: 0399039
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Abstract: F. Diaz y Diaz has discovered 99 discriminants d between $ - 3321607$ and $ - 60638515$ inclusive for which $ Q(\sqrt d )$ have a 3-rank $ {r_3} = 3$. These 99 imaginary quadratic fields are analyzed here and the class groups are given and discussed for all those of special interest. In 98 cases, the associated real quadratic fields have $ {r_3} = 2$, but for $ d = 44806173 = 3 \cdot 14935391,Q(\sqrt d )$ has a class group $ C(3) \times C(3) \times C(3)$; and this is now the smallest known d for which a real quadratic field has $ {r_3} = 3$.


References [Enhancements On Off] (What's this?)

  • [1] DANIEL SHANKS & PETER WEINBERGER, "A quadratic field of prime discriminant requiring three generators for its class group, and related theory," Acta Arith., v. 21, 1972, pp. 71-87. MR 46 #9003. MR 0309899 (46:9003)
  • [2] DANIEL SHANKS, "New types of quadratic fields having three invariants divisible by 3," J. Number Theory, v. 4, 1972, pp. 537-556. MR 47 #1775. MR 0313220 (47:1775)
  • [3] DANIEL SHANKS & RICHARD SERAFIN, "Quadratic fields with four invariants divisible by 3," Math. Comp., v. 27, 1973, pp. 183-187; "Corrigenda," ibid., p. 1012. MR 48 #8436a, b. MR 0330097 (48:8436a)
  • [4] CAROL NEILD & DANIEL SHANKS, "On the 3-rank of quadratic fields and the Euler product," Math. Comp., v. 28, 1974, pp. 279-291. MR 0352042 (50:4530)
  • [5] F. DIAZ Y DIAZ, "Sur les corps quadratiques imaginaires dont le 3-rang du groupe des classes est supérieur à 1", Séminaire Delange-Pisot-Poitou, 1973/74, no. G15.
  • [6] R. J. PORTER, "On irregular negative determinants of exponent 9n," MTAC, v. 10, 1956, pp. 22-25. MR 17, 1140. MR 0078057 (17:1140c)
  • [7] R. J. PORTER, Tables in the UMT file, MTAC, v. 7, 1953, p. 34; v. 8, 1954, pp. 96-97; v. 9, 1955, p. 26, p. 126, p. 198; v. 11, 1957, p. 275; v. 12, 1958, p. 225.
  • [8] T. CALLAHAN, "The 3-class groups of non-Galois cubic fields. I," Mathematika, v. 21, 1974, pp. 72-89. MR 0366876 (51:3122)
  • [9] T. CALLAHAN, "The 3-class groups of non-Galois cubic fields. II," Mathematika, v. 21, 1974, pp. 168-188. MR 0366876 (51:3122)
  • [10] DANIEL SHANKS, "Review of Angell's table," Math. Comp., v. 29, 1975, pp. 661-665.
  • [11] DANIEL SHANKS, "Calculation and applications of Epstein zeta functions," Math. Comp., v. 29, 1975, pp. 271-287. MR 0409357 (53:13114a)
  • [12] DAVID W. BOYD & H. KISILEVSKY, "On the exponent of the ideal class groups of complex quadratic fields," Proc. Amer. Math. Soc., v. 31, 1972, pp. 433-436. MR 44 #6644. MR 0289454 (44:6644)
  • [13] P. J. WEINBERGER, "Exponents of the class groups of complex quadratic fields," Acta Arith., v. 22, 1973, pp. 117-124. MR 47 #1776. MR 0313221 (47:1776)
  • [14] A. SCHOLZ, "Über die Beziehung der Klassenzahlen quadratischer Körper zueinander," Crelle's J., v. 166, 1932, pp. 201-203.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1976-0399039-9
Article copyright: © Copyright 1976 American Mathematical Society

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