Further irreducibility criteria for polynomials associated with the complete residue systems in any imaginary quadratic field

Let $K=\mathbb{Q}(\sqrt{m})$ be an imaginary quadratic field with $O_K$ its ring of integers. Let $\pi$ and $\beta$ be an irreducible element and a nonzero element, respectively, in $O_K$. In the authors' earlier work, it was proved for the cases, $m\not\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and $m\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ that if $\pi ={\alpha }_n{\beta }^n+{\alpha }_{n-1}{\beta }^{n-1}+\cdots +{\alpha }_1\beta +{\alpha }_0=:f(\beta )$, where $n\ge 1$, ${\alpha }_n\in O_K\setminus \{0\}$, ${\alpha }_0,\dots ,{\alpha }_{n-1}$ belong to a complete residue system modulo $\beta$, and the digits ${\alpha }_{n-1}$ and ${\alpha }_n$ satisfy certain restrictions, then the polynomial $f(x)$ is irreducible in $O_K[x]$. In this paper, we extend these results by establishing further irreducibility criteria for polynomials in $O_K[x]$. In addition, we provide elements of $\beta$ that can be applied to the new criteria but not to the previous ones.