Osnovni teorem 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.