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

For a Gaussian prime π and a nonzero Gaussian integer β=a+bi∈ℤi with a≥1 and β≥2+2, it was proved that if π=αnβn+αn−1βn−1+⋯+α1β+α0≕fβ where n≥1, αn∈ℤi\0, α0,…,αn−1 belong to a complete residue system modulo β, and the digits αn−1 and αn satisfy certain restrictions, then the polynomial fx is irreducible in ℤix. For any quadratic field K≔ℚm, it is well known that there are explicit representations for a complete residue system in K, but those of the case m≡1 mod4 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.