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