Osnovni teorem algebre

master's thesis

• Author: Nika Kišić
• Mentor: Vedran Krčadinac (mentor)
• Committee member: Vedran Krčadinac (predsjednik povjerenstva)
• Committee member: Franka Miriam Bruckler (član povjerenstva)
• Committee member: Ljiljana Arambašić (član povjerenstva)
• Committee member: Dragutin Svrtan (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-07-14, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu obradili smo nekoliko dokaza osnovnog teorema algebre. Na samom početku rada naveli smo povijesni pregled ovog teorema. U drugom smo poglavlju iskoristili svojstva kompleksnih polinoma te dali prvi dokaz. U sljedećem poglavlju smo interpretirali kompleksne polinome kao algebarske objekte. Na taj smo način mogli dokazati teorem primijenivši svojstva polja i njihovih proširenja. Posljednje poglavlje posvećeno je kompleksnoj analizi. Pokazali smo da se osnovni teorem algebre može dokazati primjenom Liouvilleova teorema. Na kraju rada izložili smo još dva dokaza koji se temelje na teoriji kompleksne analize. U prvom smo koristili nejednakost srednje vrijednosti, a u drugom princip maksimuma modula.

Abstract (english)

In this graduate thesis we prove the fundamental theorem of algebra in several ways. In the first chapter we give a historical overview of this theorem. In the second chapter we outline the first proof using properties of complex polynomials. In the next chapter, we interpret complex polynomials as algebraic objects. In this way we are able to present the second proof by applying field and extension field properties. The final chapter is dedicated to complex analysis. We demonstrate that the fundamental theorem of algebra can be proved by using Liouville’s theorem. At the end of the work, we give two additional proofs rooted in complex analysis. The first one uses the mean value inequality and the second one is based on the maximum modulus principle.

• Keywords: osnovni teorem algebre; kompleksni polinomi; algebarski objekti; kompleksna analiza; Liouvilleov teorem
• Keywords (english): fundamental theorem of algebra; complex polynomials; algebraic objects; complex analysis; Liouville's theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:799040
• 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: 2019-02-12 13:48:13