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 | References | Similar Articles | Additional Information

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**, https://doi.org/10.4064/aa-21-1-71-87**[2]**Daniel Shanks,*New types of quadratic fields having three invariants divisible by 3*, J. Number Theory**4**(1972), 537–556. MR**0313220**, https://doi.org/10.1016/0022-314X(72)90027-3**[3]**Daniel Shanks and Richard Serafin,*Quadratic fields with four invariants divisible by 3*, Math. Comp.**27**(1973), 183–187. MR**0330097**, https://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**, https://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**, https://doi.org/10.1090/S0025-5718-1956-0078057-1**[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, II*, Mathematika**21**(1974), 72–89; ibid. 21 (1974), 168–188. MR**0366876**, https://doi.org/10.1112/S0025579300005805**[9]**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**, https://doi.org/10.1112/S0025579300005805**[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.**29**(1975), 271–287. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. MR**0409357**, https://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**, https://doi.org/10.1090/S0002-9939-1972-0289454-4**[13]**P. J. Weinberger,*Exponents of the class groups of complex quadratic fields*, Acta Arith.**22**(1973), 117–124. MR**0313221**, https://doi.org/10.4064/aa-22-2-117-124**[14]**A. SCHOLZ, "Über die Beziehung der Klassenzahlen quadratischer Körper zueinander,"*Crelle's J.*, v. 166, 1932, pp. 201-203.

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DOI:
https://doi.org/10.1090/S0025-5718-1976-0399039-9

Article copyright:
© Copyright 1976
American Mathematical Society