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 ∈ ℤ[i] with a ≥ 1 and β≥2+2, it was proved that if π = αnβn + αn−1βn−1 + ⋯+α1β + α0≕f(β) where n ≥ 1, αn ∈ ℤ[i]\{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 f(x) is irreducible in ℤ[i][x]. 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.