One of my homework problems wants me explain why $x^2 +2$ is irreducible in $\mathbb{Z}\_5$. The possible roots of $x^2+2$ are $x=\pm \sqrt{-2} \notin \mathbb{Z}\_5$. Is it enough to say that since $x^2+2$ has no roots in $\mathbb{Z}\_5$, it must be irreducible in $\mathbb{Z}\_5$? Is this true for any polynomial in any number system?