Fundamental units for orders generated by a unit
Publications Mathématiques de Besançon (2015), pp. 41-68.

Soit ε une unité algébrique pour laquelle le rang du groupe des unités de l’ordre [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 [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 [ε] pour obtenir un système fondamental d’unités de cet ordre.

Let ε be an algebraic unit for which the rank of the group of units of the order [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 [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 [ε].

Reçu le : 2015-06-16
Publié le : 2016-01-24
DOI : https://doi.org/10.5802/pmb.12
Classification : 11R16,  11R27
Mots clés: Cubic unit, cubic orders, quartic unit, quartic order, fundamental units.
@article{PMB_2015____41_0,
     author = {St\'ephane R. Louboutin},
     title = {Fundamental units for orders generated by a unit},
     journal = {Publications Math\'ematiques de Besan\c con},
     publisher = {Presses universitaires de Franche-Comt\'e},
     year = {2015},
     pages = {41-68},
     doi = {10.5802/pmb.12},
     zbl = {1414.11146},
     language = {en},
     url = {pmb.centre-mersenne.org/item/PMB_2015____41_0/}
}
Stéphane R. Louboutin. Fundamental units for orders generated by a unit. Publications Mathématiques de Besançon (2015), pp. 41-68. doi : 10.5802/pmb.12. https://pmb.centre-mersenne.org/item/PMB_2015____41_0/

[BHMMS] J. Beers, D. Henshaw, C. Mccall, S. Mulay and M. Spindler, Fundamentality of a cubic unit u for [u], Math. Comp., 80 (2011), 563–578, Corrigenda and addenda, Math. Comp. 81 (2012), 2383–2387. | Article | MR 2728994 | Zbl 1280.11065

[Coh] H. Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, 138, Springer-Verlag, Berlin, 1993. | Article | Zbl 0786.11071

[LL14] J. H. Lee and S. Louboutin, On the fundamental units of some cubic orders generated by units, Acta Arith., 165 (2014), 283–299. | Article | MR 3263953 | Zbl 1307.11120

[LL15] J. H. Lee and S. Louboutin, Determination of the orders generated by a cyclic cubic unit that are Galois invariant, J. Number Theory, 148 (2015), 33–39. | Article | MR 3283165 | Zbl 1394.11073

[Lou06] S. Louboutin, The class-number one problem for some real cubic number fields with negative discriminants, J. Number Theory, 121, (2006), 30–39. | Article | MR 2268753 | Zbl 1171.11057

[Lou10] S. Louboutin, On some cubic or quartic algebraic units, J. Number Theory, 130, (2010), 956–960. | Article | MR 2600414 | Zbl 1211.11121

[Lou12] S. Louboutin, On the fundamental units of a totally real cubic order generated by a unit, Proc. Amer. Math. Soc., 140, (2012), 429–436. | Article | MR 2846312 | Zbl 1283.11152

[Lou08a] S. Louboutin, The fundamental unit of some quadratic, cubic or quartic orders, J. Ramanujan Math. Soc. 23, No. 2 (2008), 191–210. | Zbl 1165.11076

[Lou08b] S. Louboutin, Localization of the complex zeros of parametrized families of polynomials, J. Symbolic Comput., 43, (2008), 304–309. | Article | MR 2402034 | Zbl 1132.12306

[MS] S. Mulay and M. Spindler, The positive discriminant case of Nagell’s theorem for certain cubic orders, J. Number Theory, 131, (2011), 470–486. | Article | MR 2739047 | Zbl 1219.11163

[Nag] T. Nagell, Zur Theorie der kubischen Irrationalitäten, Acta Math., 55, (1930), 33–65. | Article | Zbl 56.0168.04

[PL] S.-M. Park and G.-N. Lee, The class number one problem for some totally complex quartic number fields, J. Number Theory, 129, (2009), 1338–1349. | Article | MR 2521477 | Zbl 1167.11040

[Tho] E. Thomas, Fundamental units for orders in certain cubic number fields , J. Reine Angew. Math., 310, (1979), 33-55. | Article | MR 546663 | Zbl 0427.12005

[Was] L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, 83, Springer-Verlag, Second Edition, 1997. | Article | Zbl 0966.11047