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 is a totally real cubic algebraic unit, then there exists a unit such that is a system of fundamental units of the group of the units of the cubic order , except for an infinite family for which is a square in and one sporadic exception. Not only is our proof shorter, but it enables us to prove a new result: if the conjugates and of are in , then the subgroup generated by and is of bounded index in , and if and if and are of opposite sign, then is a system of fundamental units of .

**[BHMMS]**J. Beers, D. Henshaw, C. McCall, S. Mulay and M. Spindler,

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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:
https://doi.org/10.1090/S0002-9939-2011-10924-9

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.