Let $f(x) = x^3 - 4ix^2 + 16x - (1+3i) \in (\mathbb{Z}[i])[x]$. Is $f$ irreducible over $(\mathbb{Z}[i])[x]$ ?
By Gauss Lemma, I think that the irreducibility of $f$ over $(\mathbb{Z}[i])[x]$ is the same as irreducibility over $(\mathbb{Q}[i])[x]$. For question regarding irreducibility, I think I should either apply Eisenstein or Gauss Lemma. But Eisenstein might not helpful in this case. For Gauss Lemma, I think I might need to do preliminary test like applying analogous version of Rational root test over $(\mathbb{Q}[i])[x]$ to $f$ to see if it has linear factor. Anyway, when I try to substitute some factor of $1 + 3i$ to $f(x)$, I find that it is hard to calculate if $f(x) = 0 $ or not.
Can anyone suggest a good way to detect the irreducubility of a strange polynomial like this one ?