On ℤ-Modules of Algebraic Integers

Abstract

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. (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. (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. (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. (4) There are only finitely many Pisot-cyclotomic numbers of any fixed order.