On the irreducibility of polynomials associated with the complete residue systems in any imaginary quadratic fields (Q2038316)

Summary: For a Gaussian prime \(\pi\) and a nonzero Gaussian integer \(\beta=a+bi\in\mathbb{Z}[i]\) with \(a\geq 1\) and \(|\beta|\geq 2+\sqrt{2}\), it was proved that if \(\pi= \alpha_n \beta^n+ \alpha_{n-1} \beta^{n-1}+ \cdots+ \alpha_1\beta+ \alpha_0=: f(\beta)\) where \(n\geq 1, \alpha_n\in\mathbb{Z} [i]\backslash\{0\}, \alpha_0,\dots, \alpha_{n-1}\) 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 \(\mathbb{Z} [i][x]\). For any quadratic field \(K:=\mathbb{Q} (\sqrt{m})\), it is well known that there are explicit representations for a complete residue system in \(K\), but those of the case \(m\equiv 1\pmod 4\) are inapplicable to this work. In this article, we establish a new complete residue system for such a case and then generalize the result mentioned above for the ring of integers of any imaginary quadratic field.