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].