Irreducible in $(\mathbb{Z}[i])[x]$ Prove that $p(x)=x^3-6x^2+4ix+1+3i$ is irreducible in $(\mathbb{Z}[i])[x]$. I'm not so clear on irreducibility in specific instances like this one. I know I need to show that if $p(x)=q(x)r(x)$ then W.L.O.G. $q(x)$ is a unit. Since $\mathbb{Z}[i]$ is an integral domain, it follows that $(\mathbb{Z}[i])[x]$ is also an integral domain, so the units are simply $\\{1,-1,i,-i\\}$. Some back and forth feedback would be preferred to a straightaway proof. I know that $\deg(p(x))=3$ $\Rightarrow \deg(q(x))+\deg(r(x))=3$. The question becomes how to show $\deg(q(x))=0$.

If a third degree plynomial is not irreducible, there has to be a first degree divisor. So if $x^3-6x^2-4ix+1+3i$ is not irreducible, it has a root in $\Bbb Z[i]$ and this root must divide $1+3i$. So you only have to check among the divisors of $1+3i$.