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

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.