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

$$|\mathfrak{F}(x^n-1)|\le 2$$

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

$$|\mathfrak{F}(x^n-1)|=3,4,5.$$