Abstract
Let \( \alpha \) be an algebraic number of degree \( d\geq 2 \). We consider the set \( E(\alpha) \) of positive integers \( n \) such that the primitive \( n \)th root of unity \( e^{2\pi i/n} \) is expressible as a quotient of two conjugates of \( \alpha \) over \( {} \). In particular, our results imply that \( E(\alpha) \) is small. We prove that \( |E(\alpha)|<d^{\frac{c}{\log\log d}} \), where \( c=1.04 \) for each sufficiently large \( d \). We also show that, in terms of \( d \), this estimate is best possible up to a constant, since the constant \( 1.04 \) cannot be replaced by any number smaller than \( 0.69 \).
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Funding
This research has received funding by the European Social Fund (Project 09.3.3–LMT–K–712–01–0037) under grant agreement with the Research Council of Lithuania (LMTLT).
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Translated from Sibirskii Matematicheskii Zhurnal, 2021, Vol. 62, No. 3, pp. 513–517. https://doi.org/10.33048/smzh.2021.62.303
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Dubickas, A. Cyclotomic Quotients of Two Conjugates of an Algebraic Number. Sib Math J 62, 409–412 (2021). https://doi.org/10.1134/S0037446621030034
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DOI: https://doi.org/10.1134/S0037446621030034