Abstract
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 im-ages 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 ques-tion 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 in-vestigation 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 2α0 pα1 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.
Highlights
Which rings can be homomorphic images of Z[√m ]? This question offers students an infinite number 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
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, Zn is a homomorphic image of Z[i] if and only if the prime decomposition of n is 2α0pα1 1 · · · pαkk, with α0 ∈ {0, 1}, pi ≡ 1 and αi ≥ 0 for every i ≥ 1
It seems natural to investigate the homomorphic images of Z[i]
Summary
Hierdie vraag bied aan studente ’n oneindige aantal ondersoeke (een vir elke m) wat slegs voorgraadse Wiskunde vereis. Ons gebruik die feit dat Z[i] ’n hoofideaalgebied is om te bewys dat as I = (a + bi) ’n nie-nul ideaal van Z[i] is, dan is Z[i]/I ∼= Zn vir ’n positiewe heelgetal n as en slegs as ggd{a, b} = 1, in welke geval n = a2 + b2. Deur die heelgetalle n te karakteriseer wat die vorm n = a2 + b2 het, met ggd{a, b} = 1, verkry ons die hoofresultaat van die artikel, wat beweer dat as n ≥ 2, dan is Zn ’n homomorfe beeld van Z[i] as en slegs as 2α0pα1 1 · · · pαkk die priemfaktorisering van n is, met α0 ∈ {0, 1}, pi ≡ 1 (mod 4) en αi ≥ 0 vir elke i ≥ 1. Al die liggame wat homomorfe beelde van Z[i] is, word ook bepaal
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