A Survey of Galois'' Premier Memoire

• Main Authors: Jin-Wei Shen, 沈金緯
• Other Authors: 劉瓊如
• Format: Others
• Language: en_US
• Published: 2014
• Online Access: http://ndltd.ncl.edu.tw/handle/75652380196994832961

碩士 === 國立臺灣大學 === 數學研究所 === 102 === In this paper we study the original idea of the group of substitutions of an algebraic equation with either literal or numerical coefficients. The group of substitutions of the proposed roots or, equivalently, the group of automorphisms of the minimal splitting field containing the proposed roots, is the core of the theory of algebraic equation and is one of the motivations to the development of the modern abstract group and the field theory. Although there are various publications of theory of algebraic equations and (finite) Galois theory. Our goal is the historical background of the definition of a Galois group and the difficulties of explicit constructing them, which was solved by then young Galois. The main obstacle of extending the theory from literal equations to arbitrary equations (literal or numerical) can be traced back to the work of Lagrange; and it is exactly Galois who solved the problem by his genius inventions of the Galois resolvent and the Galois group. In this paper we will inspect computation details of the algebraic solutions so that we can be fully motivated to see how subtle the definition of a Galois group is made.