Multiplicity of irreducible factors of $x^n-1 \in \mathbb{F}_q[x]$

Question

I heard a claim that $x^n-1 \in \mathbb{F}_q[x]$ factors into irreducibles as $\prod_{i=1}^k\phi_i^{p^e}$ where $n=p^en_1$ and $\gcd(n_1,p)=1$ where $p$ is the characteristic of $\mathbb{F}_q$ but I can't seem to find a reference.

It seems to me that you could use the fact that the $m-th$ cyclotomic polynomials splits into irreducibles of degree $d$ where $d$ is the order of $q$ modulo $n$ and that $\prod_{m\mid n}\Phi_m = x^n-1$? but alas, no progress

Answer 1

Note that $$x^n - 1 = (x^{n_1})^{p^e} - 1^{p^e} = (x^{n_1} - 1)^{p^e}.$$ The polynomial $x^{n_1} - 1$ is coprime to its derivative $n_1 x^{n_1 - 1}$, so it is separable and hence factors as a product of distinct irreducible polynomials.