Research History Of The Fundamental Theorem Of Algebra

The fundamental theorem of algebra is a very important and basic theorem in algebra,that is,any polynomial of degree n（n>0）has at least one root in the complex field.Its mathematical proof and historical development have always been valued by athematicians and historians of mathematics.At the same time,Gauss（Johann Carl Friedrich Gauss,1777-1855）,the core figure of this history,has also become a research focus.The four proofs he put forward in 1799^[1],1815^[2],1816^[3]and 1849^[4]contain different proving ideas respectively.Gauss’s method of proving the fundamental theorem of algebra opens up a new way to discuss the existence problem in mathematics.In view of the core role of the basic theorem of algebra in algebra and its important position in the whole mathematics,on the basis of the original literature and research literature,under the guidance of the thought of"why mathematics"^[5],this paper focuses on the reasons why there were so many proofs of basic theorems of algebra before the 20th century.An in-depth analysis of how the function is constructed in Gauss’s third proof?The main research results are as follows:（1）since the end of the 19th century,there have been abundant materials for the systematic study of the basic theorems of algebra abroad.Many scholars have studied the basic theorems from various dimensions,taking"research history"as the starting point.By comprehensively and systematically combing the historical differences and background of mathematical historians,this paper reveals the important position of Gauss’s proof in history and the change of Gauss’s position.（2）with the support of the original literature,a more detailed interpretation of the research work of the relevant mathematicians from 1746 to the end of the 19th century,combing the proof process of this theorem by some mathematicians and the motivation source of each proof,in-depth analysis of the ideas and methods of these mathematicians in the development of the theorem,reveal the ideological inheritance relationship between them,and then explain the reasons for so many proofs.（3）there are researchers abroad to restore the thought process of Gauss’s third proof.Through the full interpretation of the original literature and research literature,and on the basis of following the idea of Gauss’s original proof,this paper provides a new way to prove the function y in the third proof.