## Irreducible in F_p[x] ?

Suppose $p$ is prime and $f(x) \in \mathbb{F}_p[x]$ is monic of degree $n$. Prove that if $x^{p^n}-x=0$ in $\frac{\mathbb{F}_p[x]}{(f(x))}$ and $x^{p^{n/q}}-x\neq 0$ in $\frac{\mathbb{F}_p[x]}{(f(x))}$ for all primes $q$ dividing $n$, then $f(x)$ is irreducible in $\mathbb{F}_p[x]$.