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Gaussian periods and units in certain cyclic fields

Author: Andrew J. Lazarus
Journal: Proc. Amer. Math. Soc. 115 (1992), 961-968
MSC: Primary 11R16; Secondary 11L05, 11R27
MathSciNet review: 1093600
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Abstract: We analyze the property of period-unit integer translation (there exists a Gaussian period $ \eta $ and rational integer $ c$ such that $ \eta + c$ is a unit) in simplest quadratic, cubic, and quartic fields of arbitrary conductor. This is an extension of work of E. Lehmer, R. Schoof, and L. C. Washington for prime conductor. We also determine the Gaussian period polynomial for arbitrary conductor.

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  • [1] Paul Bachmann, Die Lehre von der Kreisteilung, Verlag and Druck von B. G. Teubner, Leipzig and Berlin, 1927.
  • [2] A. Châtelet, Arithmétique des corps abélians du troisième degré, Ann. Sci. École Norm. Sup. (4) 63 (1946), 109-160. MR 0020598 (8:568a)
  • [3] Carl Friedrich Gauss, Theoria residuorum biquadraticorum: commentatio prima, Werke, Königlichen Gesellschaft der Wissenschaften, Göttingen, 1876, pp. 65-92; originally published 1825.
  • [4] Marie-Nicole Gras, Table numérique du nombre de classes et des unités des extensions cycliques de degré 4 de $ \mathbb{Q}$, Publ. Math. Fac. Sci. Besançon, 1977/1978.
  • [5] K. Hardy, R. H. Hudson, D. Richman, K. S. Williams, and N. M. Holtz, Calculation of the class numbers of imaginary cyclic quartic fields, Math. Lecture Note Ser., vol. 7, Carleton University and Université d'Ottawa, July, 1986. MR 906194 (88m:11112)
  • [6] Helmut Hasse, Arithmetische Bestimmung von Grundeinheit und Klassenzahl in zyklischen kubischen und biquadratischen Zahlkörpern, Math. Abhand., Walter deGruyter, Berlin, 1975, pp. 285-379; originally published 1950.
  • [7] Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Math., vol. 84, Springer-Verlag, Berlin, Heidelberg, and New York, 1982. MR 661047 (83g:12001)
  • [8] Andrew J. Lazarus, Class numbers of simplest quartic fields, Number Theory (R. A. Mollin, ed.), Walter de Gruyter, Berlin and New York, 1990, pp. 313-323. MR 1106670 (92d:11119)
  • [9] -, On the class number and unit index of simplest quartic fields, Nagoya Math. J. 121 (1991), 1-13. MR 1096465 (92a:11129)
  • [10] Emma Lehmer, Connection between Gaussian periods and cyclic units, Math. Comp. 50 (1988), 535-541. MR 929551 (89h:11067a)
  • [11] Toru Nakahara, On cyclic biquadratic fields related to a problem of Hasse, Monatsh. Math. 94 (1982), 125-132. MR 678047 (84a:12007)
  • [12] René Schoof and Lawrence C. Washington, Quintic polynomials and real cyclotomic fields with large class numbers, Math. Comp. 50 (1988), 543-556. MR 929552 (89h:11067b)
  • [13] Daniel Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137-1152. MR 0352049 (50:4537)

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Keywords: Gaussian period, period polynomial, simplest fields, units
Article copyright: © Copyright 1992 American Mathematical Society

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