Fundamental units in a family of cubic fields

Veikko Ennola[1]

  • [1] Department of Mathematics University of Turku FIN-20014, Finland

Journal de Théorie des Nombres de Bordeaux (2004)

  • Volume: 16, Issue: 3, page 569-575
  • ISSN: 1246-7405

Abstract

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Let 𝒪 be the maximal order of the cubic field generated by a zero ε of x 3 + ( ℓ - 1 ) x 2 - ℓ x - 1 for ℓ ∈ ℤ , ℓ ≥ 3 . We prove that ε , ε - 1 is a fundamental pair of units for 𝒪 , if [ 𝒪 : ℤ [ ε ] ] ≤ ℓ / 3 .

How to cite

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Ennola, Veikko. "Fundamental units in a family of cubic fields." Journal de Théorie des Nombres de Bordeaux 16.3 (2004): 569-575. <http://eudml.org/doc/249260>.

@article{Ennola2004,
abstract = {Let $\mathcal\{O\}$ be the maximal order of the cubic field generated by a zero $\varepsilon $ of $x^3+(\ell -1)x^2-\ell x-1$ for $\ell \in \mathbb\{Z\}$, $\ell \ge 3$. We prove that $\varepsilon ,\varepsilon -1$ is a fundamental pair of units for $\mathcal\{O\}$, if $[\mathcal\{O\}:\mathbb\{Z\}[\varepsilon ]]\le \ell /3.$},
affiliation = {Department of Mathematics University of Turku FIN-20014, Finland},
author = {Ennola, Veikko},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {cubic field; fundamental unit; maximal order; non-abelian field; Maple computation},
language = {eng},
number = {3},
pages = {569-575},
publisher = {Université Bordeaux 1},
title = {Fundamental units in a family of cubic fields},
url = {http://eudml.org/doc/249260},
volume = {16},
year = {2004},
}

TY - JOUR
AU - Ennola, Veikko
TI - Fundamental units in a family of cubic fields
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2004
PB - Université Bordeaux 1
VL - 16
IS - 3
SP - 569
EP - 575
AB - Let $\mathcal{O}$ be the maximal order of the cubic field generated by a zero $\varepsilon $ of $x^3+(\ell -1)x^2-\ell x-1$ for $\ell \in \mathbb{Z}$, $\ell \ge 3$. We prove that $\varepsilon ,\varepsilon -1$ is a fundamental pair of units for $\mathcal{O}$, if $[\mathcal{O}:\mathbb{Z}[\varepsilon ]]\le \ell /3.$
LA - eng
KW - cubic field; fundamental unit; maximal order; non-abelian field; Maple computation
UR - http://eudml.org/doc/249260
ER -

References

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  1. B. N. Delone, D. K. Faddeev, The Theory of Irrationalities of the Third Degree. Trudy Mat. Inst. Steklov, vol. 11 (1940); English transl., Transl. Math. Monographs, vol. 10, Amer. Math. Soc., Providence, R. I., Second printing 1978. Zbl0133.30202MR4269
  2. V. Ennola, Cubic number fields with exceptional units. Computational Number Theory (A. Pethö et al., eds.), de Gruyter, Berlin, 1991, pp. 103–128. Zbl0732.11054MR1151859
  3. H. G. Grundman, Systems of fundamental units in cubic orders. J. Number Theory 50 (1995), 119–127. Zbl0828.11061MR1310739
  4. M. Mignotte, N. Tzanakis, On a family of cubics. J. Number Theory 39 (1991), 41–49, Corrigendum and addendum, 41 (1992), 128. Zbl0763.11011MR1123167
  5. E. Thomas, Fundamental units for orders in certain cubic number fields. J. Reine Angew. Math. 310 (1979), 33–55. Zbl0427.12005MR546663

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