Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



$ p$-adic computation of real quadratic class numbers

Authors: J. Buchmann, J. W. Sands and H. C. Williams
Journal: Math. Comp. 54 (1990), 855-868
MSC: Primary 11Y40; Secondary 11R29
MathSciNet review: 1010596
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathcal{K}$ be any real quadratic field and let $ {h_\mathcal{K}}$ be the class number of $ \mathcal{K}$. A method utilizing the p-adic class number formula for $ \mathcal{K}$ is described for evaluating $ {h_\mathcal{K}}$. The technique was programmed for a micro VAX II computer and run on all fields $ \mathcal{K}$ with radicand $ < {10^6}$.

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

  • [1] Y. Amice and J. Fresnel, Fonctions zêta p-adiques des corps de nombres abeliens réels, Acta Arith. 20 (1972), 353-384. MR 0337898 (49:2667)
  • [2] J. W. L. Glaisher, Residue of the product of p numbers in arithmetical progression mod $ p^2$ and $ p^3$, Messenger Math. 30 (1900-01), 71-92.
  • [3] H. W. Leopoldt, Eine p-adische Theorie der Zetawerte. II, Die p-adische T-Transformation, J. Reine Angew. Math. 274/275 (1975), 224-239. MR 0379446 (52:351)
  • [4] R. A. Mollin and H. C. Williams, Computation of the class number of a real quadratic field, Advances in the Theory of Computation and Computational Mathematics (to appear). MR 1162532 (93d:11134)
  • [5] I. S. Slavutskii, Upper bounds and numerical calculation of the number of ideal classes of real quadratic fields, Amer. Math. Soc. Transl. (2) 82 (1969), 67-72.
  • [6] R. G. Stanton, C. Sudler, Jr., and H. C. Williams, An upper bound for the period of the simple continued fraction for $ \sqrt d $, Pacific J. Math. 67 (1976), 525-536. MR 0429724 (55:2735)
  • [7] A. J. Stephens and H. C. Williams, Some computational results on a problem concerning powerful numbers, Math. Comp. 50 (1988), 619-632. MR 929558 (89d:11091)
  • [8] H. Wada, A table of ideal class numbers of real quadratic fields, Kôkyûroku in Math., no. 10, Sophia University, Tokyo, 1981.
  • [9] L. C. Washington, Introduction to cyclotomic fields, Springer-Verlag, New York, 1982. MR 718674 (85g:11001)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11Y40, 11R29

Retrieve articles in all journals with MSC: 11Y40, 11R29

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society