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Available in print format 		Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(e) ISSN 0025-5718(p)

     

Fundamentality of a cubic unit $ u$ for $ \mathbb{Z}[u]$

Author(s): J. Beers; D. Henshaw; C. K. McCall; S. B. Mulay; M. Spindler.
Journal: Math. Comp. 80 (2011), 563-578.
MSC (2010): Primary 11R16; Secondary 11R27
Posted: July 29, 2010
MathSciNet review: 2728994
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Abstract | References | Similar articles | Additional information

Abstract: Consider a cubic unit $ u$ of positive discriminant. We present a computational proof of the fact that $ u$ is a fundamental unit of the order $ \mathbb{Z}[u]$ in most cases and determine the exceptions. This extends a similar (but restrictive) result due to E. Thomas.

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H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138, Springer-Verlag (2000). MR 1228206 (94i:11105)

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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)

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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)

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S. Louboutin, On some cubic or quartic algebraic units, J. Number Theory, to appear.

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T. Nagell, Zur Theorie der kubishen Irrationalitaten, Acta Math. 55 (1930), 33-65. MR 1555314

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S.-M. Park and G.-N. Lee, The class number one problem for some totally complex quartic number fields, J. Number Theory 129 (2009), 1138-1349. MR 2521477

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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:

J. Beers
Affiliation: The College of New Jersey, Ewing, New Jersey 08628
Email: JasonBBeers@gmail.com

D. Henshaw
Affiliation: Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634
Email: davidlhenshaw@gmail.com

C. K. McCall
Affiliation: Department of Mathematics, 719 Patterson Office Tower, University of Kentucky, Lexington, Kentucky 40506-0027
Email: cmccall@ms.uky.edu

S. B. Mulay
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300
Email: mulay@math.utk.edu

M. Spindler
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Email: spindler@math.jhu.edu

DOI: 10.1090/S0025-5718-2010-02383-X
PII: S 0025-5718(2010)02383-X
Keywords: Cubic orders, fundamental units.
Received by editor(s): April 20 2009
Received by editor(s) in revised form: July 27, 2009 and October 18, 2009
Posted: July 29, 2010
Additional Notes: The authors were supported by the NSF REU award no. 0552774, 2008
Copyright of article: Copyright 2010, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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