Prsteni cijelih brojeva kvadratnih polja

master's thesis

• Title (english): Rings of integers of quadratic fields
• Author: Lucia Kurilovčan
• Mentor: Zrinka Franušić (mentor)
• Mentor: Nikola Adžaga (sumentor)
• Committee member: Zrinka Franušić (predsjednik povjerenstva)
• Committee member: Tihomir Vukelja (član povjerenstva)
• Committee member: Nikola Adžaga (član povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Committee member: Tomislav Bokulić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2024-05-02, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Neka je \(m\) cijeli broj koji nije potpuni kvadrat. Skup oblika \(\mathbb{Q}(\sqrt{m}) = \{ a+b \sqrt{m} : a,b \in \mathbb{Q} \}\), uz standardne operacije zbrajanja i množenja u \(\mathbb{C}\), čini polje i naziva se kvadratno polje. U radu se bavimo prstenima cijelih brojeva kvadratnih polja, to jest skupovima oblika \(\mathbb{Z}[\sqrt m]=\{a+b\sqrt{m}\ :\ a,b\in\mathbb{Z}\}\), za \(m \equiv 2\) ili \(3\pmod{4}\) i \(\mathbb{Z}\left[\frac{1+\sqrt m}{2}\right]=\left\{ \frac{a+b\sqrt{m}}{2} \ : \ a,b \in \mathbb{Z}, \ a \equiv b \pmod{2} \right\},\) za \(m \equiv 1\pmod{4}\), te problemom faktorizacije na ireducibilne faktore u tim prstenima.

Abstract (english)

Let \(m\) be an integer that is not a perfect square. A set of the form \(\mathbb{Q}(\sqrt{m}) = \{ a+b \sqrt{m} : a,b \in \mathbb{Q} \}\), with standard addition and multiplication in \(\mathbb{C}\), forms a field and is called a quadratic field. In the thesis, we deal with rings of integers of quadratic fields, that is, sets of the form \(\mathbb{Z}[\sqrt m]=\{a+b\sqrt{m}\ :\ a,b\in\mathbb{Z}\}\), for \(m \equiv 2\) or \(3\pmod{4}\) and \(\mathbb{Z}\left[\frac{1+\sqrt m}{2}\right]=\left\{ \frac{a+b\sqrt{m}}{2} \ : \ a,b \in \mathbb{Z}, \ a \equiv b \pmod{2} \right\},\) for \(m \equiv 1\pmod{4}\), and with the factorization problem in these rings.

• Keywords: prsteni cijelih brojeva; kvadratno polje
• Keywords (english): rings of integers; quadratic fields
• Language: croatian
• URN:NBN: urn:nbn:hr:217:356771
• Study programme: Title: Mathematics and Physics Education; specializations in: Mathematics and Physics Education; Course: Mathematics and Physics Education; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: sveučilišni magistar edukacije matematike i fizike (sveučilišni magistar edukacije matematike i fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-05-22 09:27:03