代数基本定理的证方法研究.docx

青 岛 科 技 大 学本 科 毕 业 设 计 （论 文）代数基本定理的证明方法研究题 目 ____________________________________________________________________指导教师__________________________辅导教师__________________________学生姓名__________________________学生学号__________________________102_______________________________院（部）____________________________专业________________班2014617______年 ___月 ___日代数基本定理的证明方法研究摘 要在漫长的解方程历史探索中, 数学家得出一元多次方程的解与次数关系的代数学基本定理, 代数基本定理在代数学中占有非常重要的地位。一直以来, 学者们给出了不同的方法来证明这个定理。最早的完整证明是高斯给出的纯解析方法的证明。本论文主要是对代数基本定理的已有的证明方法进行适当的归纳总结。第一章给出代数基本定理的内容并用复变函数理论中的刘维尔定理、儒歇定理、辐角原理、最大模原理、最小模原理、留数定理、柯西定理来证明代数学基本定理, 并对这些证明方法进行说明、比较与总结。第二章主要介绍了翁东东用初等方法的证明。第三章介绍了Kuhn的两个构造性的证明方法。第四章介绍了高斯的纯解析证明方法。这些证明方法都是具有代表性的证明方法。关键词：代数基本定理；复变函数；初等方法；构造性方法；解析方法STUDY ON THE PROOFS OF THE FUNDAMENTAI THEOREM OF ALGEBRAABSTRACTInthe long history of exploration in the solutions of equations,mathematicians drew a conclusion of the fundamental theorem of algebra about the relationshipbetween the roots of polynomial and the degree.Fundamental theorem of algebra plays a very important role in the algebra areas. From time being ,mathematicians have given different ways to prove this theorem.The first complete proof with pure analytical methodis given by Gauss.The main pupose of this thesis is to summarize the existingproof of the fundamental theorem of algebra. The first chapter describes the contents of the fundamental theorem of algebra and usesthe Liouville theorem, Rouche theorem, argument principle,maximum modulus principle, the minimum Modulus principle, residue theorem, Cauchys theorem of complex function theory to prove the fundamental theorem of algebra. In this chapter various proofs are described, compared and summarized. The second chapter mainly introduces the elementary method of proof of Weng Dongdong. The third chapter gives two constructive proofs of Kuhn. The fourth chapter gives pure analytic proof of Guss. These methods are all representative.KEY WORDS:The fundamental theorem of algebra; complex function; elementary methods