TITLE: Why is this polynomial irreducible over $\mathbb{Z}[i]$?
QUESTION [5 upvotes]: 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 $\mathbb{Z}[i]$?
Source.

REPLY [5 votes]: Primitive + irreducible over $\mathbb{Q}(i)$ implies irreducible over $\mathbb{Z}[i]$ by Gauss's Lemma, because $\mathbb{Z}[i]$ is a UFD with $\mathbb{Q}(i)$ as its field of fractions.
In fact, take a look at the general statement of Eisenstein's Criterion.