Factor rings of the Gaussian integers

Patterson, Cody (2004)

The original publication is available at http://www.satnt.ac.za/

Article

Whereas the homomorphic images of Z (the ring of integers) are well known, namely Z, {0} and Zn (the ring of integers modulo n), the same is not true for the homomorphic images of Z[i] (the ring of Gaussian integers). More generally, let m be any nonzero square free integer (positive or negative), and consider the integral domain Z[ √ m] = {a + b √ m | a, b ∈ Z}. Which rings can be homomorphic images of Z[ √ m]? This question offers students an infinite number (one for each m) of investigations that require only undergraduate mathematics. It is the goal of this article to offer a guide to the investigation of the possible homomorphic images of Z[ √ m] using the Gaussian integers Z[i] as an example. We use the fact that Z[i] is a principal ideal domain to prove that if I = (a+bi) is a nonzero ideal of Z[i], then Z[i]/I ∼= Zn for a positive integer n if and only if gcd{a, b} = 1, in which case n = a2 + b2. Our approach is novel in that it uses matrix techniques based on the row reduction of matrices with integer entries. By characterizing the integers n of the form n = a2 + b2, with gcd{a, b} = 1, we obtain the main result of the paper, which asserts that if n ≥ 2, then Zn is a homomorphic image of Z[i] if and only if the prime decomposition of n is 2α0pα1 1 · · · pαk k , with α0 ∈ {0, 1}, pi ≡ 1 (mod 4) and αi ≥ 0 for every i ≥ 1. All the fields which are homomorpic images of Z[i] are also determined.

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