Class groups of the quadratic fields found by F. Diaz y Diaz
Author:
Daniel Shanks
Journal:
Math. Comp. 30 (1976), 173178
MSC:
Primary 12A25; Secondary 12A50
Corrigendum:
Math. Comp. 30 (1976), 900.
Corrigendum:
Math. Comp. 30 (1976), 900.
MathSciNet review:
0399039
Fulltext PDF Free Access
Abstract 
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Abstract: F. Diaz y Diaz has discovered 99 discriminants d between and inclusive for which have a 3rank . 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 , but for has a class group ; and this is now the smallest known d for which a real quadratic field has .
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A. SCHOLZ, "Über die Beziehung der Klassenzahlen quadratischer Körper zueinander," Crelle's J., v. 166, 1932, pp. 201203.
 [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. 7187. 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. 537556. 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. 183187; "Corrigenda," ibid., p. 1012. MR 48 #8436a, b. MR 0330097 (48:8436a)
 [4]
 CAROL NEILD & DANIEL SHANKS, "On the 3rank of quadratic fields and the Euler product," Math. Comp., v. 28, 1974, pp. 279291. MR 0352042 (50:4530)
 [5]
 F. DIAZ Y DIAZ, "Sur les corps quadratiques imaginaires dont le 3rang du groupe des classes est supérieur à 1", Séminaire DelangePisotPoitou, 1973/74, no. G15.
 [6]
 R. J. PORTER, "On irregular negative determinants of exponent 9n," MTAC, v. 10, 1956, pp. 2225. 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. 9697; v. 9, 1955, p. 26, p. 126, p. 198; v. 11, 1957, p. 275; v. 12, 1958, p. 225.
 [8]
 T. CALLAHAN, "The 3class groups of nonGalois cubic fields. I," Mathematika, v. 21, 1974, pp. 7289. MR 0366876 (51:3122)
 [9]
 T. CALLAHAN, "The 3class groups of nonGalois cubic fields. II," Mathematika, v. 21, 1974, pp. 168188. MR 0366876 (51:3122)
 [10]
 DANIEL SHANKS, "Review of Angell's table," Math. Comp., v. 29, 1975, pp. 661665.
 [11]
 DANIEL SHANKS, "Calculation and applications of Epstein zeta functions," Math. Comp., v. 29, 1975, pp. 271287. 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. 433436. 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. 117124. 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. 201203.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197603990399
PII:
S 00255718(1976)03990399
Article copyright:
© Copyright 1976
American Mathematical Society
