Further factorization of $x^n-1$ over a finite field

Abstract

Let ${\mathbb{F}}_q$ be a finite field with $q$ elements and $n$ a positive integer. Martínez, Vergara and Oliveira \cite{MVO} explicitly factorized $x^n-1$ over ${\mathbb{F}}_q$ under the condition of $rad(n)|(q-1)$. In this paper, suppose that $rad(n)\nmid (q-1)$ and $rad(n)|(q^w-1)$, where $w$ is a prime, we explicitly factorize $x^n-1$ into irreducible factors in ${\mathbb{F}}_q[x]$ and count the number of its irreducible factors.