On the number of the irreducible factors of $ x^{n}-1 $ over finite fields

Let $ \mathbb{F}_q $ be the finite field of $ q $ elements, and $ \mathbb{F}_{q^{n}} $ its extension of degree $ n $. A normal basis of $ \mathbb{F}_{q^{n}} $ over $ \mathbb{F}_q $ is a basis of the form $ \{\alpha, \alpha^{q}, \cdots, \alpha^{q^{n-1}}\} $. Some problems on normal bases can be finally reduced to the determination of the irreducible factors of the polynomial $ x^{n}-1 $ in $ \mathbb{F}_q $, while the latter is closely related to the cyclotomic polynomials. Denote by $ \mathfrak{F}(x^{n}-1) $ the set of all distinct monic irreducible factors of $ x^{n}-1 $ in $ \mathbb{F}_q $. The criteria for $ |\mathfrak{F}(x^{n}-1)|\leq 2 $ have been studied in the literature. In this paper, we provide the sufficient and necessary conditions for $ |\mathfrak{F}(x^{n}-1)| = s, $ where $ s $ is a positive integer by using the properties of cyclotomic polynomials and results from the Diophantine equations. As an application, we obtain the sufficient and necessary conditions for $ |\mathfrak{F}(x^{n}-1)| = 3, 4, 5. $