Fundamental units for orders generated by a unit

Stéphane R. Louboutin

doi:10.5802/pmb.12

Stéphane R. Louboutin

Publications mathématiques de Besançon. Algèbre et théorie des nombres (2015), pp. 41-68.

Abstract
Résumé

Let $ε$ be an algebraic unit for which the rank of the group of units of the order $ℤ [ε]$ is equal to $1$ . Assume that $ε$ is not a complex root of unity. It is natural to wonder whether $ε$ is a fundamental unit of this order. It turns out that the answer is in general positive, and that a fundamental unit of this order can be explicitly given (as an explicit polynomial in $ε$ ) in the rare cases when the answer is negative. This paper is a self-contained exposition of the solution to this problem, solution which was up to now scattered in many papers in the literature. We also include the state of the art in the case that the rank of the group of units of the order $ℤ [ε]$ is greater than $1$ when now one wonders whether the set ${ε}$ can be completed in a system of fundamental units of the order $ℤ [ε]$ .

Soit $ε$ une unité algébrique pour laquelle le rang du groupe des unités de l’ordre $ℤ [ε]$ est égal à $1$ . Supposons que $ε$ ne soit pas une racine complexe de l’unité. Il est alors naturel de se demander si $ε$ est une unité fondamentale de cet ordre. Nous montrons que la réponse est en général positive et que, dans les rares cas où elle ne l’est pas, une unité fondamentale de cet ordre peut être explicitement donnée (comme polynôme en $ε$ ). Nous présentons ici une exposition complète de la solution à ce problème, solution jusqu’à présent dispersée dans plusieurs articles. Nous incluons l’état de l’art de ce problème dans le cas où la rang du groupe des unités de l’ordre $ℤ [ε]$ est strictement plus grand que $1$ , où la question naturelle est maintenant de savoir si on peut adjoindre à $ε$ d’autres unités de l’ordre $ℤ [ε]$ pour obtenir un système fondamental d’unités de cet ordre.

Received: 2015-06-16
Published online: 2016-01-24

Zbl

DOI: 10.5802/pmb.12

Classification: 11R16, 11R27
Keywords: Cubic unit, cubic orders, quartic unit, quartic order, fundamental units.

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     pages = {41--68},
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Stéphane R. Louboutin. Fundamental units for orders generated by a unit. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2015), pp. 41-68. doi : 10.5802/pmb.12. https://pmb.centre-mersenne.org/articles/10.5802/pmb.12/

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