Euklidske domene

master's thesis

• Title (english): Euclidean domains
• Author: Dora Buntić
• Mentor: Zrinka Franušić (mentor)
• Committee member: Zrinka Franušić (predsjednik povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Matko Ljulj (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2024-07-12, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Teorem o dijeljenju s ostatkom kaže da za dane cijele brojeve \(a\) i \(b \neq 0\) postoje jedinstveni cijeli brojevi \(q\) i \(r\) takvi da je \(a = bq + r\) i \(0 \leq r < |b|\). Klasa integralnih domena, to jest komutativnih prstena s jedinicom bez djelitelja nule, koji posjeduju svojstvo analogno teoremu o dijeljenju s ostatkom, nazivaju se euklidske domene. Štoviše, euklidske domene dopuštaju oblik Euklidova algoritma za određivanje najvećeg zajedničkog djelitelja. U ovom diplomskom radu pokazujemo neka svojstva euklidskih domena te primjere primjere iz klase prstena oblika \(\mathbb{Z}[\sqrt {m}]\) i \(\mathbb{Z}[\frac{1+\sqrt{m}}{2}]\).

Abstract (english)

The quotient-remainder theorem says that for given integers \(a\) and \(b \neq 0\), there exist unique integers \(q\) and \(r\) such that \(a = bq + r\) i \(0 \leq r < |b|\). A class of integer domains, i.e. commutative rings with unity that has no zero divisors, that posseses a property analogous to the quotient-remainder theorem are called Euclidean domains. Moreover, Euclidean domains admits a form of the Euclidean algorithm for calculating the greatest common divisor. In this thesis, we show some properties of Euclidean domains and give some examples in the class of rings of the form \(\mathbb{Z}[\sqrt {m}]\) and \(\mathbb{Z}[\frac{1+\sqrt{m}}{2}]\).

• Keywords: euklidske domene; klasa integralnih domena; komutativni prsteni
• Keywords (english): Euclidean domains; class of integer domains; commutative rings
• Language: croatian
• URN:NBN: urn:nbn:hr:217:677787
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar edukacije matematike (sveučilišni magistar edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-07-24 11:10:29