Osnovni teorem algebre

Abstract

U radu se proučava jedan od temeljnih teorema analize - osnovni teorem algebre. Cilj rada je prezentirati različite dokaze spomenutog teorema. Na početku, u povijesnom pregledu, navedeni su matematičari koji su se njime bavili te su spomenuti njihovi najvažniji doprinosi. Nadalje, proučava se vizualni pristup pomoću kojeg je osnovni teorem algebre ilustriran na nekoliko primjera. Kraj prvog poglavlja obuhvaća srednjoškolske teoreme vezane uz osnovni teorem algebre. Za navedene teoreme dani su strogi matematički dokazi koji se uglavnom ne navode u srednjoškolskim udžbenicima. Drugo poglavlje sastoji se od analitičkih dokaza: dokaz pomoću Liouvilleovog teorema, dokaz pomoću Velikog Picardovog teorema, dokaz pomoću Leibnizovog pravila za integrale, dokaz pomoću principa maksimuma modula za krug, dokaz razvojem u red potencija, dokaz pomoću Roucheovog teorema i dokaz temeljen na namotajnom broju. Algebarski dokaz prezentiran je u trećem poglavlju. Dokazano je da svaka kvadratna matrica ima svojstvenu vrijednost iz čega slijedi tvrdnja osnovnog teorema algebre.In this thesis we study one of the basic theorem of analysis - fundamental theorem of algebra. The goal of this paper is to present different proofs of that theorem. Firstly, in the historical overview, mathematicians who worked at this theorem are mentioned along with their most significant contributions. Furthermore, this paper study visual approach by which the fundamental theorem of algebra is illustrated with several examples. The end of the first chapter includes high school theorems connected to the fundamental theorem of algebra. For all stated theorems we included strict mathematical proofs although high school books usually do not contain their proofs. The second chapter consists of analytic proofs: proof by Liouville’s theorem, proof by Great Picard’s theorem, proof by Leibniz rule for integrals, proof by maximum modulus principle, proof by using power series, proof by Rouche’s theorem and the proof based on winding numbers. Algebraic proof is presented in third chapter. It has been proven that every square matrix has an eigenvector which implies the statement of fundamental theorem of algebra