Hiperbolički brojevi

master's thesis

• Author: Maria Crnjak
• Mentor: Zvonko Iljazović (mentor)
• Committee member: Zvonko Iljazović (predsjednik povjerenstva)
• Committee member: Mea Bombardelli (član povjerenstva)
• Committee member: Filip Najman (član povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-07-21, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U prvom smo poglavlju proučili općenito grupe i prstene, te smo definirali što je to potpuno uređeno polje. U drugom smo se poglavlju osvrnuli na prirodne, cijele i racionalne brojeve, te ih definirali unutar fiksiranog polja realnih brojeva. Zatim smo u trećem poglavlju proučili potprstene i potpolja. U četvrtom poglavlju smo definirali polje kompleksnih brojeva \(\mathbb{C}\), nakon čega smo proučavali određeno potpolje od \(\mathbb{C}\) izomorfno s \(\mathbb{R}\). Pri kraju poglavlja smo definirali morfizam uređenih skupova. U petom poglavlju smo proučavali svojstva tijela kvaterniona \(\mathbb{H}\), morfizme i izomorfizme prstenova. Također, razmotrili smo određeno polje u \(\mathbb{H}\) koje je izomorfno sa \(\mathbb{C}\). Na kraju rada, u šestom poglavlju smo definirali prsten hiperboličkih brojeva \(\mathbb{D}\) i proučili određeno polje u \(\mathbb{D}\) koje je izomorfno sa \(\mathbb{R}\).

Abstract (english)

In the first chapter, we studied groups and rings in general, and defined what a totally ordered field is. In the second chapter, we looked at natural, integer, and rational numbers, and defined them within a fixed field of real numbers. Then, in the third chapter, we studied subrings and subfields. In the fourth chapter, we defined a field of complex numbers \(\mathbb{C}\), after which we studied a certain subfield of \(\mathbb{C}\) isomorphic to \(\mathbb{R}\). At the end of the chapter, we defined morphism of ordered sets. In the fifth chapter, we studied the properties of a field of quaternions \(\mathbb{H}\), morphisms, and ring isomorphisms. We also considered a particular field in \(\mathbb{H}\) that is isomorphic to \(\mathbb{C}\). At the end of the paper, in sixth chapter, we defined a ring of hyperbolic numbers \(\mathbb{D}\) and studied a particular field in \(\mathbb{D}\) that is isomorphic to \(\mathbb{R}\).

• Keywords: grupe; prsteni; potpuno uređeno polje; hiperbolički brojevi
• Keywords (english): groups; rings; totally ordered field; hyperbolic numbers
• Language: croatian
• URN:NBN: urn:nbn:hr:217:080115
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-12-22 11:11:52