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A DYNAMICAL SYSTEM PROOF OF NIVEN’S THEOREM AND ITS EXTENSIONS

Part of: Polynomials and matrices Elementary classical functions Algebraic number theory: global fields Smooth dynamical systems: general theory

Published online by Cambridge University Press:  21 June 2023

CHATCHAWAN PANRAKSA Open the ORCID record for CHATCHAWAN PANRAKSA [Opens in a new window] ,
DETCHAT SAMART Open the ORCID record for DETCHAT SAMART [Opens in a new window]  and
SONGPON SRIWONGSA Open the ORCID record for SONGPON SRIWONGSA [Opens in a new window]
CHATCHAWAN PANRAKSA
Affiliation:
Applied Mathematics Program, Mahidol University International College, Nakhon Pathom 73170, Thailand e-mail: [email protected]
DETCHAT SAMART*
Affiliation:
Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand
SONGPON SRIWONGSA
Affiliation:
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

Niven’s theorem asserts that $\{\cos (r\pi ) \mid r\in \mathbb {Q}\}\cap \mathbb {Q}=\{0,\pm 1,\pm 1/2\}.$ In this paper, we use elementary techniques and results from arithmetic dynamics to obtain an algorithm for classifying all values in the set $\{\cos (r\pi ) \mid r\in \mathbb {Q}\}\cap K$, where K is an arbitrary number field.

Keywords

Niven’s theoremalgebraic numbersarithmetic dynamics

MSC classification

Primary: 37C25: Fixed points, periodic points, fixed-point index theory
Secondary: 11C08: Polynomials 11R04: Algebraic numbers; rings of algebraic integers 33B10: Exponential and trigonometric functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 1 , February 2024 , pp. 138 - 151
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The second author is supported by National Research Council of Thailand (NRCT) under the Research Grant for Mid-Career Scholar no. N41A640153. The third author acknowledges funding by the Office of the Permanent Secretary, Ministry of Higher Education, Science, Research and Innovation (OPS MHESI), Thailand Science Research and Innovation (TSRI) and King Mongkut’s University of Technology Thonburi (Grant No. RGNS 64-096).

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