Abstract
Let q be a prime power and let \({\mathbb {F}}_q\) be a finite field with q elements. This paper discusses the explicit factorizations of cyclotomic polynomials over \(\mathbb {F}_q\). Previously, it has been shown that to obtain the factorizations of the \(2^{n}r\)th cyclotomic polynomials, one only need to solve the factorizations of a finite number of cyclotomic polynomials. This paper shows that with an additional condition that \(q\equiv 1 \pmod p\), the result can be generalized to the \(p^{n}r\)th cyclotomic polynomials, where p is an arbitrary odd prime. Applying this result we discuss the factorization of cyclotomic polynomials over finite fields. As examples we give the explicit factorizations of the \(3^{n}\)th, \(3^{n}5\)th and \(3^{n}7\)th cyclotomic polynomials.
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Acknowledgments
The authors are very grateful to the editor and the anonymous reviewers for their valuable comments and suggestions that improved the quality of this paper. This work was supported by National Natural Science Foundation of China (Nos.11101002, 11271129 and 61370187) and Beijing Natural Science Foundation (No. 1132009). The first author was partially supported by the General program of science and technology development project of Beijing Municipal Education Commission (KM201510009013). The fourth author was partially supported by Shanghai Natural Science Foundation (12ZR1408400), and the Fundamental Research Funds for the Central Universities.
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Communicated by J. D. Key.
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Wu, H., Zhu, L., Feng, R. et al. Explicit factorizations of cyclotomic polynomials over finite fields. Des. Codes Cryptogr. 83, 197–217 (2017). https://doi.org/10.1007/s10623-016-0224-5
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DOI: https://doi.org/10.1007/s10623-016-0224-5