Base-β(C) Representations and Generalizations of Irreducibility Criteria for Polynomials over any Imaginary Quadratic Fields (European Journal of Pure and Applied Mathematics, 2025)
Let K be an imaginary quadratic field with the ring of integers O_K. In the authors' earlier work, the so-called base-β(C) representation for nonzero elements of O_K was determined, where C is a complete residue system modulo β. Using such a representation, irreducibility criteria for polynomials in O_K[x] were established. In this paper, we provide the explicit shapes of all base-β(C) representations for nonzero elements of O_K. Generalizations of such irreducibility criteria for polynomials in O_K[x] under a certain condition are also established.
Keywords: Imaginary quadratic field, Ring of integers, Complete residue system, Prime element, Irreducible polynomial