On the Irreducibility of Polynomials Associated with the Complete Residue Systems in any Imaginary Quadratic Fields

摘要

For a Gaussian prime and a nonzero Gaussian integer with and , it was proved that if where , , belong to a complete residue system modulo , and the digits and satisfy certain restrictions, then the polynomial is irreducible in . For any quadratic field , it is well known that there are explicit representations for a complete residue system in , but those of the case 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.