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

Abstract

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{mod}}\ 4) $ and $ m\equiv 1\ ({\mathrm{mod}}\ 4) $ 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 O_K\setminus\{0\} $, $ \alpha_{0}, \ldots, \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.

Keywords

• imaginary quadratic field
• ring of integers
• complete residue system
• irreducible element
• irreducible polynomial