Upper Bounds for Cyclotomic Numbers

Do Duc, Tai; Leung, Ka Hin; Schmidt, Bernhard

doi:10.5802/alco.86

Do Duc, Tai ¹; Leung, Ka Hin ²; Schmidt, Bernhard ¹

¹ Division of Mathematical Sciences School of Physical & Mathematical Sciences Nanyang Technological University Singapore 637371 Republic of Singapore
² Department of Mathematics National University of Singapore Kent Ridge, Singapore 119260 Republic of Singapore

Algebraic Combinatorics, Volume 3 (2020) no. 1, pp. 39-53.

Abstract

Let $q$ be a power of a prime $p$ , let $k$ be a nontrivial divisor of $q - 1$ and write $e = (q - 1) / k$ . We study upper bounds for cyclotomic numbers $(a, b)$ of order $e$ over the finite field $𝔽_{q}$ . A general result of our study is that $(a, b) \leq 3$ for all $a, b \in ℤ$ if $p > {(\sqrt{14})}^{k / {ord}_{k} (p)}$ . More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: $(0, 0), (0, a), (a, 0), (a, a)$ and $(a, b)$ , where $a \neq b$ and $a, b \in {1, ..., e - 1}$ . The main idea we use is to transform equations over $𝔽_{q}$ into equations over the field of complex numbers on which we have more information. A major tool for the improvements we obtain over known results is new upper bounds on the norm of cyclotomic integers.

Received: 2019-02-05
Revised: 2019-04-26
Accepted: 2019-05-08
Published online: 2020-02-12

Zbl

DOI: 10.5802/alco.86

Classification: 11T22, 11C20
Keywords: Finite fields, Cylotomic Fields, Norm Bounds

Author's affiliations:

Do Duc, Tai ¹; Leung, Ka Hin ²; Schmidt, Bernhard ¹

¹ Division of Mathematical Sciences School of Physical & Mathematical Sciences Nanyang Technological University Singapore 637371 Republic of Singapore
² Department of Mathematics National University of Singapore Kent Ridge, Singapore 119260 Republic of Singapore

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Do Duc, Tai; Leung, Ka Hin; Schmidt, Bernhard. Upper Bounds for Cyclotomic Numbers. Algebraic Combinatorics, Volume 3 (2020) no. 1, pp. 39-53. doi : 10.5802/alco.86. https://alco.centre-mersenne.org/articles/10.5802/alco.86/

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