The set of representatives and explicit factorization of xn − 1 over finite fields

Let $n$ be a positive integer and let ${𝔽}_q$ be a finite field with $q$ elements, where $q$ is a prime power and $gcd(n,q)=1$. In this paper, we give the explicit factorization of $x^n-1$ over ${𝔽}_q$ and count the number of its irreducible factors for the following conditions: $n,q$ are odd and $\text{rad}(n)|(q^2+1)$. First, we present a method to obtain the set of all representatives of $q$-cyclotomic cosets modulo $m$, where $m=gcd(n,q^2+1)$. This set of representatives is then used to find the irreducible factors of $x^n-1$ and the cyclotomic polynomial ${\mathrm{\Phi }}_n(x)$ over ${𝔽}_q$. The form of irreducible factors of $x^n-1$ is characterized such that the coefficients of these irreducible factors are followed by second-order linear recurring sequences.