伽罗瓦的代数方程思想

The exploration of Galois′ algebraic equation theory

目的探讨在代数方程根式可解性理论的发展中,伽罗瓦(Evariste Galois,1811—1832)的代数方程理论思想发展过程。方法采用历史考察与数理分析法。结果伽罗瓦是通过引进"伽罗瓦群"、"正规子群"、"置换群"等概念开始建立他的理论,并且找出了根式扩张塔和可解群之间的对应关系,利用这种对应关系最终解决了代数方程根式可解性理论这一难题。结论伽罗瓦继承了拉格朗日(J.L.Lagrange1,736—1813)问题转化的思想,并且把这一思想进行发展,使得人们对方程根式解问题的研究进入到对"结构"观念的研究,导致了抽象代数学科的诞生;伽罗瓦的研究思路是通过继承和发展前人的思想成果得出来的。

Aim To explore the theory of Galois′ algebraic equation through the depth study on the radical solution theory.Methods Historical investigation and mathematical analysis.Results Galois established his theory by introducing some concepts such as ＂Galois group＂,＂normal subgroup＂,＂permutation group＂ etc,and identifying the corresponding relation between radical expansion tower and solvable group.He finally solved the problem of algebraic equation radical solution theory.Conclusion Galois inherited the thoughts of Lagrange′s problem transforming,and he developed this thoughts so that people turned the research of algebraic equation radical solution theory into the study of structure concept.It caused the establishment of Abstract algebra discipline.His research idea originated from inheriting and developing predecessors′ achievements.