A table of fundamental pairs of units in totally real cubic fields
HTML articles powered by AMS MathViewer
- by T. W. Cusick and Lowell Schoenfeld PDF
- Math. Comp. 48 (1987), 147-158 Request permission
Abstract:
We apply a method of Cusick [5] to tabulate data on the first 250 totally real cubic fields F having discriminant $D \leqslant 6,885$. Apart from D, we list the class number H and the regulator R of F. Also given are the integer coefficients A,B,C of a defining polynomial $g(x) = {x^3} - A{x^2} + Bx - C$, its index I, and its largest zero ${R_0}$. For $j = 1,2$, we also tabulate both the integer coefficients ${X_j},{Y_j},{Z_j}$ for the two units ${E_j} = ({X_j} + {R_0}{Y_j} + R_0^2{Z_j})/I$ with norm $+ 1$, forming a fundamental pair, as well as the ${E_j}$ and the integers ${F_j} = {\text {trace}}(E_j^2)$.References
- I. O. Angell, A table of totally real cubic fields, Math. Comput. 30 (1976), no. 133, 184–187. MR 0401701, DOI 10.1090/S0025-5718-1976-0401701-6
- K. K. Billevič, On units of algebraic fields of third and fourth degree, Mat. Sb. N.S. 40(82) (1956), 123–136 (Russian). MR 0088516
- Harvey Cohn and Saul Gorn, A computation of cyclic cubic units, J. Res. Nat. Bur. Standards 59 (1957), 155–168. MR 0089868, DOI 10.6028/jres.059.016
- T. W. Cusick, Finding fundamental units in cubic fields, Math. Proc. Cambridge Philos. Soc. 92 (1982), no. 3, 385–389. MR 677463, DOI 10.1017/S0305004100060096
- T. W. Cusick, Finding fundamental units in totally real fields, Math. Proc. Cambridge Philos. Soc. 96 (1984), no. 2, 191–194. MR 757653, DOI 10.1017/S0305004100062095
- B. N. Delone and D. K. Faddeev, Theory of Irrationalities of Third Degree, Acad. Sci. URSS. Trav. Inst. Math. Stekloff, 11 (1940), 340 (Russian). MR 0004269
- 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
- H. J. Godwin, The determination of units in totally real cubic fields, Proc. Cambridge Philos. Soc. 56 (1960), 318–321. MR 117216, DOI 10.1017/s0305004100034617
- H. J. Godwin, The determination of the class-numbers of totally real cubic fields, Proc. Cambridge Philos. Soc. 57 (1961), 728–730. MR 126437, DOI 10.1017/s0305004100035854
- H. J. Godwin, A note on Cusick’s theorem on units in totally real cubic fields, Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 1, 1–2. MR 727072, DOI 10.1017/S0305004100061223
- Marie-Nicole Gras, Méthodes et algorithmes pour le calcul numérique du nombre de classes et des unités des extensions cubiques cycliques de $\textbf {Q}$, J. Reine Angew. Math. 277 (1975), 89–116 (French). MR 389845, DOI 10.1515/crll.1975.277.89
- P. Llorente and A. V. Oneto, On the real cubic fields, Math. Comp. 39 (1982), no. 160, 689–692. MR 669661, DOI 10.1090/S0025-5718-1982-0669661-3
- Michael Pohst and Hans Zassenhaus, On effective computation of fundamental units. I, Math. Comp. 38 (1982), no. 157, 275–291. MR 637307, DOI 10.1090/S0025-5718-1982-0637307-6
- Ray Steiner and Ronald Rudman, On an algorithm of Billevich for finding units in algebraic fields, Math. Comp. 30 (1976), no. 135, 598–609. MR 404204, DOI 10.1090/S0025-5718-1976-0404204-8 G. F. Voronoǐ, On a Generalization of the Algorithm for Continued Fractions (in Russian), Doctoral thesis, Warsaw, 1896.
- H. C. Williams and C. R. Zarnke, Computer calculation of units in cubic fields, Proceedings of the Second Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1972) Congressus Numerantium, No. VII, Utilitas Math., Winnipeg, Man., 1973, pp. 433–468. MR 0401705
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Math. Comp. 48 (1987), 147-158
- MSC: Primary 11R27; Secondary 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-1987-0866105-8
- MathSciNet review: 866105