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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the fundamental units of a totally real cubic order generated by a unit
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by Stéphane R. Louboutin PDF
Proc. Amer. Math. Soc. 140 (2012), 429-436 Request permission

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 $\eta \in \textbf {Z}[\epsilon ]$ such that $\{\epsilon ,\eta \}$ is a system of fundamental units of the group $U_\epsilon$ of the units of the cubic order $\textbf {Z}[\epsilon ]$, except for an infinite family for which $\epsilon$ is a square in $\textbf {Z}[\epsilon ]$ 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 $\textbf {Z}[\epsilon ]$, 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 $U_\epsilon$.
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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
  • Email: stephane.louboutin@univmed.fr
  • 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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 429-436
  • MSC (2010): Primary 11R16, 11R27
  • DOI: https://doi.org/10.1090/S0002-9939-2011-10924-9
  • MathSciNet review: 2846312