Abel Equation On The Contribution

This thesis, with the history of algebraic equation as the thread, the inner logic analysis of mathematics as the principle method, through the investigation on the original literature and regarding works, attempts to interpret clearly the idea of Abel's proof on the algebraical unsolvability of the quintic equation, and to have an analysis on the beginnings of the idea in Abel's paper so as to expose his thought wholly and systematically.This thesis will be divided into three parts for the above-mentioned target:In order to lay the foundation of the related knowledge and determine the background of the problem, the first part is to introduce and to show in detail the significant theories in the development of algebraic equation before Abel, especially to explain the fundamental theorem of symmetric polynomial and the conception of solvable equation in radicals.The second part is to study and analyze the works and ideas from several important mathematicians for providing the critical conceptions and ideas which are adopted in Abel's work. Lagrange initiated his work by investigating the known methods of solving the algebraic equations of the degree less than five, through which he summarized out a general method for solving equations and proposed his celebrated idea of'resolvent'. Ruffini, inheriting Lagrange's thought in permutation, was actually the first to publish a proof of the unsolvability. And his idea was generalized by Cauchy into the theorem of permutation which afterwards was used in Abel's proof towards unsolvability.The Last part is the focal point of this thesis which center on the procedure and the idea of the paper published in 1824 by Abel which gave a complete proof of that for the quintic equation the algebraic solution is impossible. Abel uses the method of'reductio ad absurdum': he assumes that the quintic equation is solvable at the very beginning, which achieves contradictory results at the end and disconfirms the assumption.