All irreducible factors of $p(x)=x^{p^n}-x \in \mathbb{F}_p[x]$ have degrees that divide $n$

Question

I want to prove that if I write $p(x)=x^{p^n}-x \in \mathbb{F}_p[x]$ as a product of irreducible polynomials, then the degree of each of them divides $n$.
I have already proved that any irreducible polynomial in $\mathbb{F}_p[x]$ whose degree is a divisor of $n$ divides $p(x)$. However I am stuck here. I have tried proving the main statement by reduction to absurd by assuming that $$p(x)=q_1(x)\cdot \ldots \cdot q_s(x)$$ where, for example, $l = \deg(q_1(x))$ and $l$ doesn't divide $n$, but I haven't been able to reach a contradiction. Can someone help me?