Fakorringe van die Gauss-heelgetalle
| dc.contributor.advisor | Smith, Kirby C. | en_ZA |
| dc.contributor.author | Patterson, Cody | en_ZA |
| dc.date.accessioned | 2012-02-13T12:56:42Z | |
| dc.date.available | 2012-02-13T12:56:42Z | |
| dc.date.issued | 2004 | |
| dc.description | The original publication is available at http://www.satnt.ac.za/ | en_ZA |
| dc.description | CITATION: Smith, K. C. & Patterson, C. 2004. Fakorringe van die Gauss-heelgetalle. Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie, 23(4):a201, doi:10.4102/satnt.v23i4.201. | |
| dc.description.abstract | ENGLISH ABSTRACT: Factor rings of the Gaussian integers 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. | en_ZA |
| dc.description.abstract | AFRIKAANSE OPSOMMING: In teenstelling met die faktorringe van Z (die ring van heelgetalle) wat goed bekend is, naamlik Z, {0} en Zn (die ring van heelgetalle modulo n), is dieselfde nie waar vir die homomorfe beelde van Z[i] (die ring van Gauss-heelgetalle) nie. Meer algemeen, laat m enige nie-nul kwadraatvrye heelgetal (positief of negatief) wees, en beskou die integraal- gebied Z[√m ] = {a + b√m | a, b ∈ Z}. Watter ringe is homomorfe beelde van Z[√m ]? Hierdie vraag bied aan studente ’n oneindige aantal ondersoeke (een vir elke m) wat slegs voorgraadse Wiskunde vereis. Die doel van hierdie artikel is om as ’n riglyn tot die bepal- ing van die homomorfe beelde van Z[√m ] te dien deur die Gauss-heelgetalle Z[i] as ’n voorbeeld te gebruik. 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 2 2 n as en slegs as ggd{a, b} = 1, in welke geval n = a + b . Ons benadering is oorspronklik in die sin dat dit matrikstegnieke gebruik wat gebaseer is op ry-reduksie van matrikse met heeltallige inskrywings. 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 2α0 pα1 ··· pαk die priemfak-torisering 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. | af_ZA |
| dc.description.version | Publishers' Version | en_ZA |
| dc.format.extent | 12 pages | |
| dc.identifier.citation | Patterson, C. and Kirby, C.S. 2004. Factor rings of the Gaussian integers. Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie, 23(4), 114-125. | en_ZA |
| dc.identifier.issn | 0254-3486 (print) | |
| dc.identifier.issn | 2222-4173 (online) | |
| dc.identifier.uri | http://hdl.handle.net/10019.1/19736 | |
| dc.language.iso | af | en_ZA |
| dc.publisher | AOSIS | en_ZA |
| dc.rights.holder | Copyright is retained by the author(s) | en_ZA |
| dc.subject.lcsh | Factor rings | en_ZA |
| dc.subject.lcsh | Gaussian integers | en_ZA |
| dc.subject.lcsh | Rings of integers | en_ZA |
| dc.subject.lcsh | Rings (Algebra) | en_ZA |
| dc.subject.lcsh | Homomorphisms (Mathematics) | en_ZA |
| dc.title | Fakorringe van die Gauss-heelgetalle | af_ZA |
| dc.type | Article | en_ZA |