Explicit factorization of $x^n-1\in {\mathbb{F}}_q[x]$

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 $x^{2t}-a{x}^t+b$ and explicit splitting of $x^n-1$ into irreducible factors is given.

Submitted to Designs, Codes and Cryptography. 9 pages