Let $f=X^7-(7-6i)X^3+5X^2+3+6i\in\mathbb{Z}[i][X]$. Check whether $f$ is irreducible:
- over $\mathbb{Z}[i]$
- over $\mathbb{Q}(i)$
Probably I will have to use Einstein criterion with some substitution, but I have problems with notation: what do $\mathbb{Z}[i]$ and $\mathbb{Q}(i)$ mean?
$$\begin{align} (1+4i)&(1+2i)=-7+6i\\ (1-2i)&(1+2i)=5\\ 3&(1+2i)=3+6i \end{align}$$ Now $1+2i$ doesn't divide $3$ as $|1+2i|=5>3$ and $$ (a+ib)(c+id)=(1+2i)\implies(ac-bd,bc+ad)=(1,2)\implies (a^2+b^2)(c^2+d^2)=5\implies a^2+b^2=1, c^2+d^2=5\;(\text{WLOG})\implies a+ib=\pm1,\pm i $$ So $a+ib$ is a unit in $\mathbb{Z}[i]$ and $1+2i$ is a prime in $\mathbb{Z}[i]$. So applying Eisenstein's criterion we see $f$ is indeed irreducible over $\mathbb{Z}[i]$.