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]

Read more

Summary

Watter ringe is homomorfe beelde van

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

Omdat die rye van
SUMMARY
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call