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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
Corrigendum:
Math. Comp. 30 (1976), 900.
Corrigendum:
Math. Comp. 30 (1976), 900.
MathSciNet review:
0399039
Full-text PDF Free Access
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Abstract: F. Diaz y Diaz has discovered 99 discriminants d between and inclusive for which have a 3-rank . 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 .
- [1]
Daniel
Shanks and Peter
Weinberger, A quadratic field of prime discriminant requiring three
generators for its class group, and related theory, Acta Arith.
21 (1972), 71–87. MR 0309899
(46 #9003)
- [2]
Daniel
Shanks, New types of quadratic fields having three invariants
divisible by 3, J. Number Theory 4 (1972),
537–556. MR 0313220
(47 #1775)
- [3]
Daniel
Shanks and Richard
Serafin, Quadratic fields with four invariants
divisible by 3, Math. Comp. 27 (1973), 183–187. MR 0330097
(48 #8436a), http://dx.doi.org/10.1090/S0025-5718-1973-0330097-0
- [4]
Carol
Neild and Daniel
Shanks, On the 3-rank of quadratic fields and
the Euler product, Math. Comp. 28 (1974), 279–291. MR 0352042
(50 #4530), http://dx.doi.org/10.1090/S0025-5718-1974-0352042-5
- [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 9_{𝑛}, Math. Tables Aids
Comput. 10 (1956),
22–25. MR
0078057 (17,1140c), http://dx.doi.org/10.1090/S0025-5718-1956-0078057-1
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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,
II, Mathematika 21 (1974), 72–89; ibid. 21
(1974), 168–188. MR 0366876
(51 #3122)
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T.
Callahan, The 3-class groups of non-Galois cubic fields. I,
II, Mathematika 21 (1974), 72–89; ibid. 21
(1974), 168–188. MR 0366876
(51 #3122)
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DANIEL SHANKS, "Review of Angell's table," Math. Comp., v. 29, 1975, pp. 661-665.
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Daniel
Shanks, Calculation and applications of
Epstein zeta functions, Math. Comp. 29 (1975), 271–287.
Collection of articles dedicated to Derrick Henry Lehmer on the occasion of
his seventieth birthday. MR 0409357
(53 #13114a), http://dx.doi.org/10.1090/S0025-5718-1975-0409357-2
- [12]
David
W. Boyd and H.
Kisilevsky, On the exponent of the ideal class
groups of complex quadratic fields., Proc.
Amer. Math. Soc. 31
(1972), 433–436. MR 0289454
(44 #6644), http://dx.doi.org/10.1090/S0002-9939-1972-0289454-4
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P.
J. Weinberger, Exponents of the class groups of complex quadratic
fields, Acta Arith. 22 (1973), 117–124. MR 0313221
(47 #1776)
- [14]
A. SCHOLZ, "Über die Beziehung der Klassenzahlen quadratischer Körper zueinander," Crelle's J., v. 166, 1932, pp. 201-203.
- [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:
http://dx.doi.org/10.1090/S0025-5718-1976-0399039-9
PII:
S 0025-5718(1976)0399039-9
Article copyright:
© Copyright 1976 American Mathematical Society
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