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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
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.
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 109 , Issue 1 , February 2024 , pp. 138 - 151
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
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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