Further factorization of $x^n-1$ over finite fields (II)

TL;DR

This paper explicitly factors $x^n-1$ over finite fields and counts irreducible factors when the order of $q$ modulo $rad(n)$ is a product of two primes, extending previous results to new cases.

Contribution

It provides explicit factorization and counting formulas for $x^n-1$ over finite fields when the order of $q$ modulo $rad(n)$ is a product of two primes, a case not previously fully addressed.

Findings

Explicit factorization of $x^n-1$ over finite fields for specific cases.

Counting formulas for irreducible factors when the order is a product of two primes.

Extension of previous results to new algebraic cases.

Abstract

Let $\Bbb F_q$ be a finite field with $q$ elements. Let $n$ be a positive integer with radical $rad(n)$, namely, the product of distinct prime divisors of $n$. If the order of $q$ modulo $rad(n)$ is either 1 or a prime, then the irreducible factorization and a counting formula of irreducible factors of $x^n-1$ over $\Bbb F_q$ were obtained by Mart\'{\i}nez, Vergara, and Oliveira (Des Codes Cryptogr 77 (1) : 277-286, 2015) and Wu, Yue, and Fan (Finite Fields Appl 54: 197-215, 2018). In this paper, we explicitly factorize $x^{n}-1$ into irreducible factors in $\Bbb F_q[x]$ and calculate the number of the irreducible factors when the order of $q$ modulo $rad(n)$ is a product of two primes.