戴德金的伽罗瓦群成因探析

On the Factors of Establishment of Dedekind’s Galois Group

伽罗瓦理论与伽罗瓦建立的理论的重要区别在于前者关注群与域两种代数结构的对应关系,后者侧重于用置换群研究代数方程的根式可解性。戴德金的工作是这一历史转折的重要环节。戴德金发现方程的伽罗瓦群是方程所在域的特有属性,将伽罗瓦群理解为域扩张的自同构群,突出了伽罗瓦理论中群和域两个主要概念之间的相互作用,初步勾勒出现代伽罗瓦基本定理。基于拉格朗日和伽罗瓦的结果,对戴德金的相关工作进行梳理和分析,可以使我们深刻地理解戴德金的伽罗瓦群之思想来源及形成过程。

The important difference between the modern Galois Theory and the theory established by Galois is that the former focuses on the correspondence between the algebraic structure of the group and the field,while the latter focuses on the radical solvability of the algebraic equation.Dedekind’s work on Galois group is an important part of this historical transformation.Dedekind found that the Galois group of the equation is a unique property of the field of the equation,and using the isomorphism of the field to deal with the Galois Theory,the Galois group is defined as the automorphism group of the field,originally outlining the basic theorem of modern Galois Theory.Based on the results of Lagrange and Galois,this paper makes a systematic pectination and analysis of Dedekind’s related works,which is helpful to deeply understand the origin and formation process of Dedekind’s Galois group.