Konačna polja

master's thesis

• Author: Iva Kriste
• Mentor: Vedran Krčadinac (mentor)
• Committee member: Vedran Krčadinac (predsjednik povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Committee member: Mirko Primc (član povjerenstva)
• Committee member: Damir Bakić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-02-24, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu izneseni su osnovni pojmovi i teoremi vezani za konačna polja. Na samom početku definirani su prsteni, kao jedni od osnovnih algebarskih struktura, i sve vezano uz njih, uključujući polja. Objašnjeno je što su to prsteni polinoma i kvocijentni prsteni koji su bili potrebni za konstrukciju konačnih polja. Primjerima je pokazano na koji način možemo konstruirati konačno polje bilo kojeg reda oblika \(p^n\), s pomoću ireducibilnog polinoma stupnja \(n\) nad \(\mathbb{F}^p\). Koristeći pojam polja razlaganja dokazali smo egzistenciju i jedinstvenost konačnog polja reda \(p^n\). Osim toga dokazali smo Wedderburnov teorem, za koji smo definirali ciklotomijske polinome.

Abstract (english)

In this thesis we present the main concepts and results which we need to define finite fields. At the beginning we define rings, as one of the fundamental algebraic structures and everything related to them, including fields. We define polynomials and residue class rings, which are required for the construction of finite fields. In examples we show how to construct a finite field of any order \(p^n\) from a irreducible polynomial of degree \(n\) over \(\mathbb{F}^p\). Using the concept of splitting field, we prove the existence and uniqueness of the field of order \(p^n\). In addition, we prove Wedderburn’s theorem for which we define cyclotomic polynomials.

• Keywords: konačna polja; Wedderburnov teorem
• Keywords (english): finite fields; Wedderburn’s theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:328803
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2017-04-26 12:22:08