Class numbers of the simplest cubic fields

Washington, Lawrence C.

doi:10.1090/S0025-5718-1987-0866122-8

Class numbers of the simplest cubic fields

Author: Lawrence C. Washington
Journal: Math. Comp. 48 (1987), 371-384
MSC: Primary 11R16; Secondary 11R20, 14K07
DOI: https://doi.org/10.1090/S0025-5718-1987-0866122-8
MathSciNet review: 866122
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Using the "simplest cubic fields" of D. Shanks, we give a modified proof and an extension of a result of Uchida, showing how to obtain cyclic cubic fields with class number divisible by n, for any n. Using 2-descents on elliptic curves, we obtain precise information on the 2-Sylow subgroups of the class groups of these fields. A theorem of H. Heilbronn associates a set of quartic fields to the class group. We show how to obtain these fields via these elliptic curves.

[1] William W. Adams and Michael J. Razar, Multiples of points on elliptic curves and continued fractions, Proc. London Math. Soc. (3) 41 (1980), no. 3, 481–498. MR 591651, https://doi.org/10.1112/plms/s3-41.3.481
[2] Armand Brumer and Kenneth Kramer, The rank of elliptic curves, Duke Math. J. 44 (1977), no. 4, 715–743. MR 457453
[3] Algebraic number theory, Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union. Edited by J. W. S. Cassels and A. Fröhlich, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR 0215665
[4] Harvey Cohn, A device for generating fields of even class number, Proc. Amer. Math. Soc. 7 (1956), 595–598. MR 79613, https://doi.org/10.1090/S0002-9939-1956-0079613-9
[5] T. W. Cusick, Lower bounds for regulators, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 63–73. MR 756083, https://doi.org/10.1007/BFb0099441
[6] H. Eisenbeis, G. Frey, and B. Ommerborn, Computation of the 2-rank of pure cubic fields, Math. Comp. 32 (1978), no. 142, 559–569. MR 480416, https://doi.org/10.1090/S0025-5718-1978-0480416-4
[7] H. Heilbronn, On the 2-classgroup of cubic fields, Studies in Pure Mathematics (Presented to Richard Rado), Academic Press, London, 1971, pp. 117–119. MR 0280461
[8] J.-F. Mestre, Courbes elliptiques et formules explicites, Seminar on number theory, Paris 1981–82 (Paris, 1981/1982) Progr. Math., vol. 38, Birkhäuser Boston, Boston, MA, 1983, pp. 179–187 (French). MR 729167
[9] J.-F. Mestre, Groupes de classes d’ideaux non cycliques de corps de nombres, Seminar on number theory, Paris 1981–82 (Paris, 1981/1982) Progr. Math., vol. 38, Birkhäuser Boston, Boston, MA, 1983, pp. 189–200 (French). MR 729168
[10] Daniel Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137–1152. MR 352049, https://doi.org/10.1090/S0025-5718-1974-0352049-8
[11] J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1975, pp. 33–52. Lecture Notes in Math., Vol. 476. MR 0393039
[12] Kôji Uchida, Class numbers of cubic cyclic fields, J. Math. Soc. Japan 26 (1974), 447–453. MR 360518, https://doi.org/10.2969/jmsj/02630447
[13] Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674
[14] H. Weber, Lehrbuch der Algebra, vol. I, 3rd ed., 1898; reprinted, Chelsea, New York, 1961.