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On the computation of a table of complex cubic fields with discriminant $ D>-10\sp 6$


Authors: Gilbert W. Fung and H. C. Williams
Journal: Math. Comp. 55 (1990), 313-325
MSC: Primary 11R16; Secondary 11Y40
DOI: https://doi.org/10.1090/S0025-5718-1990-1023760-X
Erratum: Math. Comp. 63 (1994), 433.
Erratum: Math. Comp. 63 (1994), 433.
MathSciNet review: 1023760
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Abstract: A method for finding all the nonisomorphic complex cubic fields with discriminant $ D > - {10^6}$ is described. Three different methods were used to find the class number of each of these fields. The speed of these techniques is discussed and several tables illustrating the computational results are presented. These include tables of the distribution of the fields and the class numbers and the class group structures of these fields.


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

  • [1] I. O. Angell, A table of complex cubic fields, Bull. London Math. Soc. 51 (1973), 37-38. MR 0318099 (47:6648)
  • [2] P. Barrucand, J. Loxton, and H. C. Williams, Some explicit upper bounds on the class number and regulator of a cubic field with negative discriminant, Pacific J. Math. 128 (1987), 209-222. MR 888515 (88e:11107)
  • [3] P. Barrucand, H. C. Williams, and L. Baniuk, A computational technique for determining the class number of a pure cubic field, Math. Comp. 30 (1976), 312-323. MR 0392913 (52:13726)
  • [4] J. Buchmann and H. C. Williams, On the computation of the class number of an algebraic number field, Math. Comp. 53 (1989), 679-688. MR 979937 (90a:11128)
  • [5] H. Cohen and J. Martinet, Class groups of number fields: numerical heuristics, Math. Comp. 48 (1987), 123-138. MR 866103 (88e:11112)
  • [6] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II. Proc. Roy. Soc. London Ser. A 322 (1971), 405-420. MR 0491593 (58:10816)
  • [7] B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree, Transl. Amer. Math. Soc., vol. 10, Amer. Math. Soc., Providence, R.I., 1964. MR 0160744 (28:3955)
  • [8] V. Ennola and R. Turunen, On totally real cubic fields, Math. Comp. 44 (1985), 495-518. MR 777281 (86e:11100)
  • [9] P. Llorente and E. Nart, Effective determination of the decomposition of the rational primes in a pure cubic field, Proc. Amer. Math. Soc. 87 (1983), 579-585. MR 687621 (84d:12003)
  • [10] P. Llorente and J. Quer, On totally real cubic fields with discriminant $ D < {10^7}$, Math. Comp. 50 (1988), 581-594. MR 929555 (89g:11099)
  • [11] D. Shanks, A survey of quadratic, cubic and quartic algebraic number fields (from a computational point of view), Cong. Numer. 17 (1976), 15-40. MR 0453691 (56:11951)
  • [12] -, Review of [1], Math. Comp. 29 (1975), 661-665.
  • [13] H. C. Williams, Continued fractions and number-theoretic computations, Rocky Mountain J. Math. 15 (1985), 621-655. MR 823273 (87h:11129)
  • [14] -, Effective primality tests for some integers of the form $ A{5^n} - 1$ and $ A{7^n} - 1$, Math. Comp. 48 (1987), 385-403. MR 866123 (88b:11089)
  • [15] H. C. Williams, G. Cormack, and E. Seah, Calculation of the regulator of a pure cubic field, Math. Comp. 34 (1980), 567-611. MR 559205 (81d:12003)
  • [16] H. C. Williams and C. R. Zarnke, Some algorithms for solving a cubic congruence modulo p, Utilitas Math. 6 (1974), 285-306. MR 0389730 (52:10561)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1990-1023760-X
Article copyright: © Copyright 1990 American Mathematical Society

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