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 + b i ∈ ℤ i with a ≥ 1 and β ≥ 2 + 2 , it was proved that if π = α n β n + α n − 1 β n − 1 + ⋯ + α 1 β + α 0 ≕ f β where n ≥ 1 , α n ∈ ℤ i \ 0 , α 0 , … , α n − 1 belong to a complete residue system modulo β , and the digits α n − 1 and α n satisfy certain restrictions, then the polynomial f x is irreducible in ℤ i x . For any quadratic field K ≔ ℚ m , it is well known that there are explicit representations for a complete residue system in K , but those of the case m ≡ 1 mod 4 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.

A characterization of Sophie Germain primes

Let [Formula: see text] be an odd integer. It is shown that [Formula: see text] is a complete residue system modulo [Formula: see text] for some permutation [Formula: see text] of [Formula: see text] if and only if [Formula: see text] is a Sophie Germain prime. Partial results are obtained also for the case [Formula: see text] even.

Linear Congruence Relation and Complete Residue Systems

Linear Congruence Relation and Complete Residue Systems In this paper, we defined the congruence relation and proved its fundamental properties on the base of some useful theorems. Then we proved the existence of solution and the number of incongruent solution to a linear congruence and the linear congruent equation class, in particular, we proved the Chinese Remainder Theorem. Finally, we defined the complete residue system and proved its fundamental properties.

Complete residue systems in the quadratic domainZ(e2πi/3)

Several representations for a complete residue system in the Euclidean domainZ(ω)are presented in this paper.

Complete residue systems in the ring of matrices of rational integers

This paper deals with the characterizations of the complete residue systemmod. G, whereGis anyn×nmatrix, in the ring ofn×nmatrices.