On the computation of a table of complex cubic fields with discriminant $D>-10^ 6$
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- by Gilbert W. Fung and H. C. Williams PDF
- Math. Comp. 55 (1990), 313-325 Request permission
Erratum: Math. Comp. 63 (1994), 433.
Erratum: Math. Comp. 63 (1994), 433.
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
- I. O. Angell, A table of complex cubic fields, Bull. London Math. Soc. 5 (1973), 37–38. MR 318099, DOI 10.1112/blms/5.1.37
- Pierre Barrucand, John 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), no. 2, 209–222. MR 888515, DOI 10.2140/pjm.1987.128.209
- Pierre Barrucand, H. C. Williams, and L. Baniuk, A computational technique for determining the class number of a pure cubic field, Math. Comp. 30 (1976), no. 134, 312–323. MR 392913, DOI 10.1090/S0025-5718-1976-0392913-9
- Johannes Buchmann and H. C. Williams, On the computation of the class number of an algebraic number field, Math. Comp. 53 (1989), no. 188, 679–688. MR 979937, DOI 10.1090/S0025-5718-1989-0979937-4
- H. Cohen and J. Martinet, Class groups of number fields: numerical heuristics, Math. Comp. 48 (1987), no. 177, 123–137. MR 866103, DOI 10.1090/S0025-5718-1987-0866103-4
- H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405–420. MR 491593, DOI 10.1098/rspa.1971.0075
- B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence, R.I., 1964. MR 0160744
- Veikko Ennola and Reino Turunen, On totally real cubic fields, Math. Comp. 44 (1985), no. 170, 495–518. MR 777281, DOI 10.1090/S0025-5718-1985-0777281-8
- Pascual Llorente and Enric Nart, Effective determination of the decomposition of the rational primes in a cubic field, Proc. Amer. Math. Soc. 87 (1983), no. 4, 579–585. MR 687621, DOI 10.1090/S0002-9939-1983-0687621-6
- Pascual Llorente and Jordi Quer, On totally real cubic fields with discriminant $D<10^7$, Math. Comp. 50 (1988), no. 182, 581–594. MR 929555, DOI 10.1090/S0025-5718-1988-0929555-8
- Daniel Shanks, A survey of quadratic, cubic and quartic algebraic number fields (from a computational point of view), Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory, and Computing (Louisiana State Univ., Baton Rouge, La., 1976) Congressus Numerantium, No. XVII, Utilitas Math., Winnipeg, Man., 1976, pp. 15–40. MR 0453691 —, Review of [1], Math. Comp. 29 (1975), 661-665.
- Hugh C. Williams, Continued fractions and number-theoretic computations, Rocky Mountain J. Math. 15 (1985), no. 2, 621–655. Number theory (Winnipeg, Man., 1983). MR 823273, DOI 10.1216/RMJ-1985-15-2-621
- H. C. Williams, Effective primality tests for some integers of the forms $A5^n-1$ and $A7^n-1$, Math. Comp. 48 (1987), no. 177, 385–403. MR 866123, DOI 10.1090/S0025-5718-1987-0866123-X
- H. C. Williams, G. Cormack, and E. Seah, Calculation of the regulator of a pure cubic field, Math. Comp. 34 (1980), no. 150, 567–611. MR 559205, DOI 10.1090/S0025-5718-1980-0559205-7
- H. C. Williams and C. R. Zarnke, Some algorithms for solving a cubic congruence modulo $p$, Utilitas Math. 6 (1974), 285–306. MR 389730
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Math. Comp. 55 (1990), 313-325
- MSC: Primary 11R16; Secondary 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-1990-1023760-X
- MathSciNet review: 1023760