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

TL;DR

This paper extends the explicit factorization of $x^n - 1$ over finite fields to cases where the radical of $n$ divides $q^w - 1$ but not $q - 1$, providing new factorization formulas and counts.

Contribution

It generalizes previous factorization results to new conditions involving prime powers, offering explicit formulas and counts for irreducible factors.

Findings

Explicit factorization of $x^n - 1$ over $F_q$ under new divisibility conditions

Derived formulas for counting irreducible factors

Extended understanding of polynomial factorization over finite fields

Abstract

Let $\Bbb F_q$ be a finite field with $q$ elements and $n$ a positive integer. Mart\'inez, Vergara and Oliveira \cite{MVO} explicitly factorized $x^{n} - 1$ over $\Bbb F_q$ 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 $\Bbb F_q[x]$ and count the number of its irreducible factors.