On Z-Modules of Algebraic Integers

Bell, J. P. ; Hare, K. G.

Canadian journal of mathematics, Tome 61 (2009) no. 2, pp. 264-281

Résumé

Let $q$ be an algebraic integer of degree $d\ge 2$ . Consider the rank of the multiplicative subgroup of ${{\mathbb{C}}^{*}}$ generated by the conjugates of $q$ . We say $q$ is of full rank if either the rank is $d-1$ and $q$ has norm $\pm 1$ , or the rank is $d$ . In this paper we study some properties of $\mathbb{Z}[q]$ where $q$ is an algebraic integer of full rank. The special cases of when $q$ is a Pisot number and when $q$ is a Pisot-cyclotomic number are also studied. There are four main results. (1) If $q$ is an algebraic integer of full rank and $n$ is a fixed positive integer, then there are only finitely many $m$ such that $\text{disc}\left( \mathbb{Z}\left[ {{q}^{m}} \right] \right)=\text{disc}\left( \mathbb{Z}\left[ {{q}^{n}} \right] \right)$ . (2) If $q$ and $r$ are algebraic integers of degree $d$ of full rank and $\mathbb{Z}[{{q}^{n}}]=\mathbb{Z}[{{r}^{n}}]$ for infinitely many $n$ , then either $q=\omega {r}'$ or $q=\text{Norm}{{(r)}^{2/d}}\omega /r'$ , where $r'$ is some conjugate of $r$ and $\omega $ is some root of unity. (3) Let $r$ be an algebraic integer of degree at most 3. Then there are at most 40 Pisot numbers $q$ such that $\mathbb{Z}[q]=\mathbb{Z}[r]$ . (4) There are only finitely many Pisot-cyclotomic numbers of any fixed order.

Détail

DOI : 10.4153/CJM-2009-013-9

Mots-clés : 11R04, 11R06, algebraic integers, Pisot numbers, full rank, discriminant

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