Research On The History Of Galois Theory

The establishment of Galois theory has brought great innovations on algebra research which about targets,contents and methods.In order to solve the two problems of why Galois theory can be established and why it can be transformed from the theory about algebraic equations to the general theory about abstract algebraic structures,based on the original literature and research literature,taking Galois group,the core concept of the theory as the clue,this dissertation focuses on the works of Galois and Dedekind.The historical development of Galois theory from Galois to Dedekind is systematically studied by investigating the research motivation,the internal relationship of key ideas,and legacy problems of core mathematicians,and the main results are as follows:1.The pre history of Galois theory is sorted out.Lagrange's route-map formulates the general method of solving algebraic equations,implies two clues of permutation group and number field,and depicts the rudiment of Galois fundamental theorem.Through the irreducible concept,Gauss used the concept of field in theory about algebraic equations for the first time.The solution of cyclotomic equation is reduced to the decomposition of the equation into the product of simple factors by adding a series of radicals of prime degree to the known field.Taking Lagrange as the starting point,Ruffini and Cauchy developed the permutation theory.On the basis of Gauss,Abel strengthened the ideas of field extension,and he defined the solvable of an algebraic equation by radicals as its coefficient field is expanded into its split field by finite radical extension of prime order.These works provide an important ideological source of the establishment and development of Galois theory.2.The proofs of Galois' s main theorems are recovered from Galois' s own idiolect and terminology,and the logical chain of evolution of algebraically solvable theory of algebraic equations among Lagrange,Gauss and Galois is revealed.The Gauss' period is another expression of Lagrange's permutation set which keeps the resolvent invariable in cyclotomic equation.Galois' s Galois group of equation is the general form of Gauss' period in theory of algebraic equation.With the introduction of Galois group of equation and its normal subgroup,Galois solves the legacy problems of Lagrange,extends Gauss' theory about cyclotomic equation to the general situation,and establishes the Galois theory about algebraic equation.Galois is influenced by Gauss both in the use of concepts and in the overall ideas.Gauss' theory of circular equation is the main source of Galois' s thought.3.The meaning and usage of terms such as “permutation” in the development of Galois theory are clarified,and the difficulties and doubts in understanding caused by terminology is solved,and it has found a problem according Galois' s idiolect and terminology which was ignored for many years in his last letter.The problem is analyzed in this dissertation.4.Dedekind's Eine Vorlesung über Algebra is systematically interpreted,several unexplored specific contributions in his work are discovered,and Dedekind's development of Galois' s theory of algebraic equations is discussed.Under the research goal of establishing the general theory of algebraic integers,Dedekind's treatment for Galois theory in 1850s was not centered on the algebraic equation.He took the Lagrange route-map as the basis of Galois' s algebraic equation theory,and then successfully extracted the core corresponding to groups and fields from Galois' s theory.This work provided the ideological source of his later abstract development of Galois theory.5.The development of Dedekind on Galois theory in the 1890s are researched.Dedekind took the relationship between number fields as the subject of algebraic research,and adopted the method of linear algebra and introduced isomorphic mapping to establish the internal connection between number fields,which made him obtain that Galois group is an automorphism group of field extension and all the details of Galois fundamental theorem in the number field,and his work also contains a method to calculate Galois group by extending mappings.By analyzing and comparing the differences and connections between Dedekind's Galois theory and the work of Weber and Artin,it shows the profound influence of Dedekind on Weber and the ideological inheritance between Dedekind and Artin.The historical development of Galois theory from Galois to Dedekind is not only an important process of the establishment and generalization of Galois theory,but also an important transition from algebraic equation to algebraic number field in the research center of algebra.Galois established Galois theory of algebraic equations on the basis of Lagrange and Gauss;Dedekind made essential contributions to the transformation of Galois theory of algebraic equations to modern Galois theory.