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1. CJM 2012 (vol 64 pp. 254)

Bell, Jason P.; Hare, Kevin G.
 Corrigendum to On $\mathbb{Z}$-modules of Algebraic Integers'' We fix a mistake in the proof of Theorem 1.6 in the paper in the title. Keywords:Pisot numbers, algebraic integers, number rings, Schmidt subspace theoremCategories:11R04, 11R06

2. CJM 2009 (vol 61 pp. 264)

Bell, J. P.; Hare, K. G.
 On $\BbZ$-Modules of Algebraic Integers Let $q$ be an algebraic integer of degree $d \geq 2$. Consider the rank of the multiplicative subgroup of $\BbC^*$ generated by the conjugates of $q$. We say $q$ is of {\em 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 $\BbZ[q]$ 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. \begin{compactenum}[\rm(1)] \item 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 $\disc\left(\BbZ[q^m]\right)=\disc\left(\BbZ[q^n]\right)$. \item If $q$ and $r$ are algebraic integers of degree $d$ of full rank and $\BbZ[q^n] = \BbZ[r^n]$ for infinitely many $n$, then either $q = \omega r'$ or $q={\rm Norm}(r)^{2/d}\omega/r'$, where $r'$ is some conjugate of $r$ and $\omega$ is some root of unity. \item Let $r$ be an algebraic integer of degree at most $3$. Then there are at most $40$ Pisot numbers $q$ such that $\BbZ[q] = \BbZ[r]$. \item There are only finitely many Pisot-cyclotomic numbers of any fixed order. \end{compactenum} Keywords:algebraic integers, Pisot numbers, full rank, discriminantCategories:11R04, 11R06

3. CJM 2008 (vol 60 pp. 1267)

Blake, Ian F.; Murty, V. Kumar; Xu, Guangwu
 Nonadjacent Radix-$\tau$ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields In his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-$\tau$ expansion of integers in the number fields $\Q(\sqrt{-3})$ and $\Q(\sqrt{-7})$. The (window) nonadjacent form of $\tau$-expansion of integers in $\Q(\sqrt{-7})$ was first investigated by Solinas. For integers in $\Q(\sqrt{-3})$, the nonadjacent form and the window nonadjacent form of the $\tau$-expansion were studied. These are used for efficient point multiplications on Koblitz curves. In this paper, we complete the picture by producing the (window) nonadjacent radix-$\tau$ expansions for integers in all Euclidean imaginary quadratic number fields. Keywords:algebraic integer, radix expression, window nonadjacent expansion, algorithm, point multiplication of elliptic curves, cryptographyCategories:11A63, 11R04, 11Y16, 11Y40, 14G50
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