There is a passage on the crazy project saying $x^3+12x^2+18x+6$ is irreducible over $\mathbb{Z}[i]$.
I'm trying to use Eisenstien's Criterion to figure it out. I know that 3 is irreducible in $\mathbb{Z}[i]$, so since this polynomial is 3-Eisenstein, that would mean it's irreducible over $\mathbb{Q}[i]$, right? Then why is it also irreducible over $\mathbb{Z}[i]$?