Explicit factorization of $x^n-1\in \mathbb F_q[x]$

F. E. Brochero Martínez; C. R. Giraldo Vergara; L. Batista de Oliveira

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arXiv · 1404.6281

Explicit factorization of $x^n-1\in \mathbb F_q[x]$

F. E. Brochero Martínez·C. R. Giraldo Vergara·L. Batista de Oliveira

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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 $x^n-1$ 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 is given.

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F. E. Brochero Martínez, C. R. Giraldo Vergara, L. Batista de Oliveira. 2014-05-18. Explicit factorization of $x^n-1\in \mathbb F_q[x]$. https://arxiv.org/abs/1404.6281

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