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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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$.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10924-9
PII: S 0002-9939(2011)10924-9
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.