Mod P Irreducibility Test

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Problem 56 (Mod P Irreducibility Test)

Let $p$ be a prime and suppose that $f(x)\in \mathbb{Z}[x]$. Let $\bar f (x)$ be the polynomial in $\mathbb{Z}_p[x]$ obtained by reducing the coefficients of $f(x)$ modulo $p$. Prove that if if $\bar f (x)$ is irreducible over $\mathbb{Z}_p$ and $\text{deg }\bar f(x) = \text{deg }f(x)$, then $f(x)$ is irreducible over $\mathbb{Q}$.

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