Abstract
Let \(\mathbb {F}_q\) be a finite field and \(n\) a positive integer. In this article, we prove that, under some conditions on \(q\) and \(n\), the polynomial \(x^n-1\in \mathbb {F}_q[x]\) can be split into irreducible binomials \(x^t-a\) and an explicit factorization into irreducible factors is given. Finally, weakening one of our hypothesis, we also obtain factors of the form \(x^{2t}-ax^t+b\) and explicit splitting of \(x^n-1\) into irreducible factors.
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Acknowledgments
We would like to thank the anonymous referees for the comments about the first version of this article; in particular, by proposing the question about the converse of Theorem 1.
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Communicated by R. Hill.
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Brochero Martínez, F.E., Giraldo Vergara, C.R. & de Oliveira, L.B. Explicit factorization of \(x^n-1\in \mathbb {F}_q[x]\) . Des. Codes Cryptogr. 77, 277–286 (2015). https://doi.org/10.1007/s10623-014-0005-y
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DOI: https://doi.org/10.1007/s10623-014-0005-y