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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
DOI: https://doi.org/10.1090/S0002-9939-2011-10924-9
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 $\eta \in \textbf {Z}[\epsilon ]$ 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

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.