Abstract
Recently the authors [12] showed that the algebraic integers of the form \({-m+\zeta_k}\) are bases of a canonical number system of \({\mathbb {Z}[{\zeta_k]}}\) provided \({m \geqq \phi(k)+1}\), where \({\zeta_k}\) denotes a k-th primitive root of unity and \({\phi}\) is Euler’s totient function. In this paper we are interested in the questions whether two bases \({-m+\zeta_k}\) and \({-n+\zeta_k}\) are multiplicatively independent. We show the multiplicative independence in case that 0 < |m−n| < 106 and |m|, |n| > 1.
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V. Ziegler was supported by the Austrian Science Fund (FWF) under the project P 24801-N26.
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Madritsch, M.G., Ziegler, V. On multiplicatively independent bases in cyclotomic number fields. Acta Math. Hungar. 146, 224–239 (2015). https://doi.org/10.1007/s10474-015-0500-2
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DOI: https://doi.org/10.1007/s10474-015-0500-2