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On the fundamental units of a totally real cubic order generated by a unit

Author: Stéphane R. Louboutin
Journal: Proc. Amer. Math. Soc. 140 (2012), 429-436
MSC (2010): Primary 11R16, 11R27
Published electronically: June 9, 2011
MathSciNet review: 2846312
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Abstract: We give a new and short proof of J. Beers, D. Henshaw, C. McCall, S. Mulay and M. Spindler following a recent result: if $ \epsilon$ is a totally real cubic algebraic unit, then there exists a unit $ \eta\in {\bf Z}[\epsilon]$ such that $ \{\epsilon,\eta\}$ is a system of fundamental units of the group $ U_\epsilon$ of the units of the cubic order $ {\bf Z}[\epsilon]$, except for an infinite family for which $ \epsilon$ is a square in $ {\bf Z}[\epsilon]$ and one sporadic exception. Not only is our proof shorter, but it enables us to prove a new result: if the conjugates $ \epsilon'$ and $ \epsilon''$ of $ \epsilon$ are in $ {\bf Z}[\epsilon]$, then the subgroup generated by $ \epsilon$ and $ \epsilon'$ is of bounded index in $ U_\epsilon$, and if $ \epsilon >1>\vert\epsilon ' \vert\geq\vert\epsilon'' \vert>0$ and if $ \epsilon'$ and $ \epsilon''$ are of opposite sign, then $ \{\epsilon,\epsilon'\}$ is a system of fundamental units of $ U_\epsilon$.

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

  • [BHMMS] J. Beers, D. Henshaw, C. McCall, S. Mulay and M. Spindler,
    Fundamentality of a cubic unit $ u$ for $ {\bf Z}[u]$.
    Math. Comp. 80 (2011), 563-578. MR 2728994
  • [Cus] T. W. Cusik.
    Lower bounds for regulators.
    Lecture Notes in Math. 1068 (1984), 63-73. MR 756083 (85k:11052)
  • [Lou02] S. Louboutin.
    The exponent three class group problem for some real cyclic cubic number fields.
    Proc. Amer. Math. Soc. 130 (2002), 353-361. MR 1862112 (2002h:11106)
  • [Lou06] S. Louboutin.
    The class-number one problem for some real cubic number fields with negative discriminants.
    J. Number Theory 121 (2006), 30-39. MR 2268753 (2007k:11189)
  • [Lou08] S. Louboutin,
    The fundamental unit of some quadratic, cubic or quartic orders.
    J. Ramanujan Math. Soc. 23, No.2 (2008), 191-210. MR 2432797 (2009h:11175)
  • [Lou10] S. Louboutin,
    On some cubic or quartic algebraic units.
    J. Number Theory 130 (2010), 956-960. MR 2600414 (2011b:11156)
  • [Nag] T. Nagell.
    Zur Theorie der kubischen Irrationalitäten.
    Acta Math. 55 (1930), 33-65. MR 1555314
  • [PL] S.-M. Park and G.-N. Lee.
    The class number one problem for some totally complex quartic number fields.
    J. Number Theory 129 (2009), 1338-1349. MR 2521477 (2010d:11126)
  • [Tho] E. Thomas.
    Fundamental units for orders in certain cubic number fields.
    J. Reine Angew. Math. 310 (1979), 33-55. MR 546663 (81b:12009)

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Additional Information

Stéphane R. Louboutin
Affiliation: Institut de Mathématiques de Luminy, UMR 6206, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France

Keywords: Fundamental units, cubic orders
Received by editor(s): June 15, 2010
Received by editor(s) in revised form: September 21, 2010, and November 24, 2010
Published electronically: June 9, 2011
Dedicated: Dedicated to Florence F.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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