God forgave my poor English... Mathematics is a subject characterized by abstraction; group theory is especially anadvancement of this characteristic. In the 250 years after finding the solution for cubic equations andbiquadratic equations and beforethe nineteenth century, mathematicians, including many of the most greatminds of human history i.e. I. Newton, L. Euler, J. L. Lagrange and J. K. F.Gauss, were all feverishly trying to find the solution for the quintic functions. However, the workwere eventually pushed to a right way by three tragic guys,Italian P. Ruffini , Norwegian N. H. Abel andFrench mathematician E. Galois, they proved that normal quintic functions have no rootsolutions. E. Galois, the most talented and youngest dead mathematician, created thetheory of groups and raised our study level of mathematic from the relationbetween roots and coefficients of the numeric equation to a new stageof the relations between sets and operations, and it emphasized theimportance of symmetry of the elements in the set. The solvability of the quintic equations then become related tothe symmetric properties of the group-theory-described polynomial of degree n.We will discuss this latter. What progress had been made in the 250 years of exploration before group theory? (作为一个数学渣渣，在此坦诚敬告，以上以下都是个人顺理成章化的理解，除了人名，其它不一定靠谱哟~ 正版详解，良心推荐《从一元一次方程到伽罗瓦理论》 冯承天) From the solution of quadratic equation, namely the Vieta's formulas( 韦达公式 ) ， people have long found the closerelation between roots and coefficients of the equation. French mathematicianA. Girard gave out the relation in terms of polynomial of power n.One step further, Newton first understood that the polynomials are'symmetric polynomials', in other words, they are invariant under thepermutation of the roots, which can be easily understood by writing theequation into multiple multiplication term.(see PPT). So he developed theconcept of primary symmetric polynomial, and gave out the Newton'stheorem: Any polynomial of variables a1,a2,...,an, can be uniquely written as apolynomial of the primary polynomials of a1,a2,...,an. The most important thing here, I think, is the concept of symmetric polynomials and the discovery of permutation invariability of the roots.
题目:群论 主讲人:周池春 时间: 2015年4月2日 星期四下午 4:30 地点: 16教学楼308室 引言: 群论在数学和物理以及化学中的重要性不言而喻。因此做理论研究，掌握群论知识是必不可少的。然而在学习群论时，抽象的符号，以及晦涩的定义让初学者较为头疼。 本次讨论班，是一个关于群论的基础性和介绍性的讲解。我将给大家简单的介绍一下群论中包含的一些基本概念，于此同时简单的介绍一下群表示的一些内容。整个讨论班中，我将大量的利用例子，帮助大家直观的掌握群论的基本概念，与一些基本定理。希望对大家以后的学习有所帮助。 内容: 1.代数简单介绍。 2.群的基本概念。 3.群表示的一些介绍。 参考文献 1 The Theory of Group Characters and Matrix Representations of Groups,DUDLEY E.LITTLEWOOD 2 Morton Hamermesh-Group theory and its application to physical problems 3 物理学中的群论.马中骐 4 物理学中的群论(上册).陶瑞宝
标签: 群论
