Show that polynomial $f\left( x \right) = x\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)\left( {x - 4} \right) - a$ is irreducible in $\mathbb{Q}$, where $a \in - 2 + 5\mathbb{Z}$.
I have shown that $f$ doesn't have zeroes in $\mathbb{Q}$. We also know that since $f$ is primitive, by Gauss lemma, it is irreducible in $\mathbb{Q}$ iff it is irreducible in $\mathbb{Z}$.
I figured we could use $\pi :\mathbb{Z}\left[ X \right] \to \left( {\mathbb{Z}/5\mathbb{Z}} \right)\left[ X \right]$, show that it is a homomorphism and that irreducibility of $\pi \left( f \right)$ implies irreducibility of $f$.
But, assuming all these statements are true, which I haven't bothered yet to check, how can we see that $\left( {\pi \left( f \right)} \right)\left( x \right) = {x^5} + 4x + 2$ is irreducible in $\left( {\mathbb{Z}/5\mathbb{Z}} \right)\left[ X \right]$?