$x^3-9$ is irreducible over $\mathbb{Z}$ I am trying to prove that $x^3-9$ is irreducible over $\mathbb{Z}$. The approach I usually try is Eisenstein's criterion, and the fact that $f\in\mathbb{Z}[X]$ irreducible $\iff$ $f(mx+n)\in\mathbb{Z}[X]$ irreducible. The thing is, every value I plug in seems to fail the condition "$p^2$ does not divide $a_0$". I looked at $f(x\pm1),f(x\pm 2),f(x\pm3)$. The other thing I tried is $x^3-9$ will be irreducible if it is irreducible in $\mathbb{F}_p[X]$ for some $p$. Again, small values don't work. Apparently $p=31$ does the job, but that feels disproportionate. Am I missing something?

It's much easier than that, as it's degree $3$, then it should have a linear factor (the only ways to factor it is as $3$ degree $1$ polynomials or $1$ degree $1$ and $1$ degree two), as it doesn't have a root in $\Bbb Z$ (use the rational root theorem), it's irreducible.