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Distinct unit generated totally complex quartic fields

Authors: Lajos Hajdu and Volker Ziegler
Journal: Math. Comp. 83 (2014), 1495-1512
MSC (2010): Primary 11R16, 11R27, 11D85
Published electronically: July 30, 2013
MathSciNet review: 3167469
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Abstract: The problem of characterization of rings whose elements can be expressed as sums of their units has a long history and is also of current interest. In this paper we take up the question of describing totally complex quartic number fields $ K$ with the property that every algebraic integer in $ K$ is the sum of distinct units of $ K$. In particular, we give a short list containing all such fields.

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

  • [1] Paul Belcher, A test for integers being sums of distinct units applied to cubic fields, J. London Math. Soc. (2) 12 (1975/76), no. 2, 141-148. MR 0409409 (53 #13164)
  • [2] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235-265. Computational algebra and number theory (London, 1993). MR 1484478,
  • [3] Takeo Funakura, On integral bases of pure quartic fields, Math. J. Okayama Univ. 26 (1984), 27-41. MR 779772 (86c:11089)
  • [4] Bernard Jacobson, Sums of distinct divisors and sums of distinct units, Proc. Amer. Math. Soc. 15 (1964), 179-183. MR 0160746 (28 #3957)
  • [5] Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723 (95f:11085)
  • [6] Ken Nakamula, Certain quartic fields with small regulators, J. Number Theory 57 (1996), no. 1, 1-21. MR 1378570 (97h:11128),
  • [7] Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, PWN--Polish Scientific Publishers, Warsaw, 1974. Monografie Matematyczne, Tom 57. MR 0347767 (50 #268)
  • [8] J. Śliwa, Sums of distinct units, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 11-13 (English, with Russian summary). MR 0345929 (49 #10658)
  • [9] The PARI Group, Bordeaux.
    PARI/GP, version 2.1.5, 2004.
    available from
  • [10] Jörg Thuswaldner and Volker Ziegler, On linear combinations of units with bounded coefficients, Mathematika 57 (2011), no. 2, 247-262. MR 2825236,
  • [11] Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575 (97h:11130)

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Additional Information

Lajos Hajdu
Affiliation: University of Debrecen, Institute of Mathematics H-4010 Debrecen, P.O. Box 12. Hungary

Volker Ziegler
Affiliation: Institute for Analysis and Computational Number Theory, Graz University of Technology Steyrergasse 30/IV, A-8010 Graz, Austria

Keywords: Unit sum number, additive unit structure, digit expansions
Received by editor(s): May 24, 2012
Received by editor(s) in revised form: August 30, 2012
Published electronically: July 30, 2013
Additional Notes: The first author’s research was supported in part by the OTKA grants K75566, K100339, NK101680, and by the TÁMOP 4.2.1./B-09/1/KONV-2010-0007 project. The project is implemented through the New Hungary Development Plan, cofinanced by the European Social Fund and the European Regional Development Fund.
The second author’s research was supported by the Austrian Science Found (FWF) under the project J2886-NT
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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