Explicit factorization of xⁿ-1∈ \mathbb F_q[x]

2014/04/24 by Martínez, F. E. Brochero, Vergara, C. R. Giraldo, de Oliveira, L. Batista
#12E05(primary) and 94B05(secondary) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1404.6281

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 xⁿ-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 xⁿ-1 into irreducible factors is given.

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