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Quintic polynomials and real cyclotomic fields with large class numbers

Authors: René Schoof and Lawrence C. Washington
Journal: Math. Comp. 50 (1988), 543-556
MSC: Primary 11R11; Secondary 11R16, 11R21, 11R27
MathSciNet review: 929552
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Abstract: We study a family of quintic polynomials discoverd by Emma Lehmer. We show that the roots are fundamental units for the corresponding quintic fields. These fields have large class numbers and several examples are calculated. As a consequence, we show that for the prime $ p = 641491$ the class number of the maximal real subfield of the pth cyclotomic field is divisible by the prime 1566401. In an appendix we give a characterization of the "simplest" quadratic, cubic and quartic fields.

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  • [1] M. Abramowitz & I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1964.
  • [2] J. W. S. Cassels, An Introduction to the Geometry of Numbers, Springer-Verlag, New York, 1971. MR 0306130 (46:5257)
  • [3] T. W. Cusick, "Lower bounds for regulators," in Number Theory, Noordwijkerhout 1983, Proceedings of the Journées Arithmétiques, Lecture Notes in Math., vol. 1068, Springer-Verlag, New York, 1984. MR 756083 (85k:11052)
  • [4] G. Cornell & L. C. Washington, "Class numbers of cyclotomic fields," J. Number Theory, v. 21, 1985, pp. 260-274. MR 814005 (87d:11079)
  • [5] G. Gras & M.-N. Gras, "Calcul du nombre de classes et des unités des extensions abéliennes réelles de Q," Bull. Sci. Math., v. 101, 1977, pp. 97-129. MR 0480423 (58:586)
  • [6] J. C. Lagarias, H. W. Lenstra & C. P. Schnorr, "Korkine-Zolotarev bases and successive minima of a lattice and its reciprocal lattice," submitted to Combinatorica.
  • [7] E. Lehmer, "Connection between Gaussian periods and cyclic units," Math. Comp., v. 50, 1988, pp. 535-541. MR 929551 (89h:11067a)
  • [8] C. Moser, "Nombre de classes d'une extension cyclique réelle de Q, de degré 4 ou 6 et de conducteur premier," Math. Nachr., v. 102, 1981, pp. 45-52. MR 642140 (83m:12012)
  • [9] C. Moser & J.-J. Payan, "Majoration du nombre de classes d'un corps cubique cyclique de conducteur premier," J. Math. Soc. Japan, v. 33, 1981, pp. 701-706. MR 630633 (83a:12006)
  • [10] M. Pohst, "Regulatorabschätzungen für total reelle algebraische Zahlkörper," J. Number Theory, v. 9, 1977, pp. 459-492. MR 0460274 (57:268)
  • [11] E. Seah, L. C. Washington & H. C. Williams, "The calculation of a large cubic class number with an application to real cyclotomic fields," Math. Comp., v. 41, 1983, pp. 303-305. MR 701641 (84m:12008)
  • [12] D. Shanks, Solved and Unsolved Problems in Number Theory, 3rd ed., Chelsea, New York, 1985. MR 798284 (86j:11001)
  • [13] D. Shanks, "The simplest cubic fields," Math. Comp., v. 28, 1974, pp. 1137-1152. MR 0352049 (50:4537)
  • [14] F. Van der Linden, "Class numbers of real cyclotomic fields," Math. Comp., v. 39, 1982, pp. 693-707. MR 669662 (84e:12005)
  • [15] L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Math., vol. 83, Springer-Verlag, New York, 1982. MR 718674 (85g:11001)
  • [16] H. C. Williams & J. Broere, "A computational technique for evaluating $ L(1,\chi )$ and the class number of a real quadratic field," Math. Comp., v. 30, 1976, pp. 887-893. MR 0414522 (54:2623)

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Keywords: Cyclotomic fields, class number, unit group, geometry of numbers
Article copyright: © Copyright 1988 American Mathematical Society

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