Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Class numbers of the simplest cubic fields

Author: Lawrence C. Washington
Journal: Math. Comp. 48 (1987), 371-384
MSC: Primary 11R16; Secondary 11R20, 14K07
MathSciNet review: 866122
Full-text PDF

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.

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

  • [1] W. Adams & M. Razar, "Multiples of points on elliptic curves and continued fractions," Proc. London Math. Soc., v. 41, 1980, pp. 481-498. MR 591651 (82c:14031)
  • [2] A. Brumer & K. Kramer, "The rank of elliptic curves," Duke Math. J., v. 44, 1977, pp. 715-743. MR 0457453 (56:15658)
  • [3] J. W. S. Cassels & A. Fröhlich, Algebraic Number Theory, Thompson Book Co., Washington, D. C., 1967. MR 0215665 (35:6500)
  • [4] H. Cohn, "A device for generating fields of even class number," Proc. Amer. Math. Soc., v. 7, 1956, pp. 595-598. MR 0079613 (18:114b)
  • [5] T. Cusick, "Lower bounds for regulators," Number Theory (Noordwijkerhout, 1983), Lecture Notes in Math., vol. 1068, Springer-Verlag, Berlin and New York, 1984, pp. 63-73. MR 756083 (85k:11052)
  • [6] H. Eisenbeis, G. Frey & B. Ommerborn, "Computation of the 2-rank of pure cubic fields," Math. Comp., v. 32, 1978, pp. 559-569. MR 0480416 (58:579)
  • [7] H. Heilbronn, "On the 2-classgroup of cubic fields," in Studies in Pure Mathematics (L. Mirsky, ed.), Academic Press, New York, 1971, pp. 117-119. MR 0280461 (43:6181)
  • [8] J. F. Mestre, "Courbes elliptiques et formules explicites," Séminaire Théorie des Nombres, Grenoble, 1982. MR 729167 (85g:14025)
  • [9] J. F. Mestre, "Groupes de classes d'idéaux non cycliques de corps de nombres," Séminaire Théorie des Nombres, Paris, 1981-1982, Birkhäuser, Boston-Basel-Stuttgart, 1983, pp. 189-200. MR 729168 (85m:11066)
  • [10] D. Shanks, "The simplest cubic fields," Math. Comp., v. 28, 1974, pp. 1137-1152. MR 0352049 (50:4537)
  • [11] J. Tate, "Algorithm for determining the type of a singular fiber in an elliptic pencil," Modular Functions of One Variable IV, Lecture Notes in Math., vol. 476, Springer-Verlag, Berlin and New York, 1975, pp. 33-52. MR 0393039 (52:13850)
  • [12] K. Uchida, "Class numbers of cubic cyclic fields," J. Math. Soc. Japan, v. 26, 1974, pp. 447-453. MR 0360518 (50:12966)
  • [13] L. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York-Heidelberg-Berlin, 1982. MR 718674 (85g:11001)
  • [14] H. Weber, Lehrbuch der Algebra, vol. I, 3rd ed., 1898; reprinted, Chelsea, New York, 1961.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11R16, 11R20, 14K07

Retrieve articles in all journals with MSC: 11R16, 11R20, 14K07

Additional Information

Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society