Upotpunjenja uređenih polja

master's thesis

• Title (english): Completions of edited fields
• Author: Barbara Pedić
• Mentor: Zvonko Iljazović (mentor)
• Committee member: Zvonko Iljazović (predsjednik povjerenstva)
• Committee member: Franka Miriam Brückler (član povjerenstva)
• Committee member: Željka Milin Šipuš (član povjerenstva)
• Committee member: Marko Erceg (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-03-04, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu, kroz prvo poglavlje proučavani su pojmovi uređene grupe, uređenih prstena i polja. Definirao se pojam niza te opisala konvergencija nizova u uređenim prstenima. U drugom se poglavlju proučavalo Cauchyjeve nizove u uređenim poljima. Uveden je pojam relacije ekvivalencije na skupu svih Cauchyjevih nizova u uređenom polju te pripadni kvocijentni skup. Definirane su binarne operacije \(\oplus \) i \(\odot\) na skupu \(\mathcal{C}(P,+,\cdot, \leqslant)/_{\thicksim}\) te dokazano da je \( (\mathcal{C}(P,+,\cdot, \leqslant)/_{\thicksim}, \oplus, \odot) \) polje. Na kraju je definirana relacija uređaja \(\preceq\) na \(\mathcal{C}(P,+,\cdot, \leqslant)/_{\thicksim}\) te je dokazano da je \( (\mathcal{C}(P,+,\cdot, \leqslant)/_{\thicksim}, \oplus, \odot, \preceq) \) uređeno polje.

Abstract (english)

In this thesis through first chapter we studied ordered groups, ordered rings and fields. The concept of sequence was defined and we described convergence of sequence in ordered fields. In second chapter we studied Cauchy sequences in ordered fields. We introduced an equivalence relation on the set of all Cauchy sequences in an ordered field and a corresponding quotient set. We defined binary operations \(\oplus \) and \(\odot\) on the set \(\mathcal{C}(P,+,\cdot, \leqslant)/_{\thicksim}\) and proved that (\( (\mathcal{C}(P,+,\cdot, \leqslant)/_{\thicksim}, \oplus, \odot) \) is a field. Finally, we defined an order \(\preceq\) na \(\mathcal{C}(P,+,\cdot, \leqslant)/_{\thicksim}\) and we proved that je \( (\mathcal{C}(P,+,\cdot, \leqslant)/_{\thicksim}, \oplus, \odot, \preceq) \)is an ordered field.

• Keywords: uređena grupa; uređeni prsten; uređeno polje; Cauchyjevi nizovi
• Keywords (english): ordered groups; ordered rings; ordered fields; Cauchy sequences
• Language: croatian
• URN:NBN: urn:nbn:hr:217:082688
• 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: 2021-03-25 12:28:29