The Early Historical Research On Class Field Theory

Class field theory is the theory of abelian extension on number fields,and it was a significant accomplishment in algebraic number theory in the 20 th century.Starting with Kronecker’s conjecture in 1853,the theory reached peak and were broadly generalized in the 1930 s thanks to efforts by such mathematicians as Hilbert,Furtw(?)ngler,Takagi Teiji and Artin.Based on the study of a large number of related original literature and research literature,coupled with methods of conceptual analysis,chronicle and literature analysis,this dissertation takes the early history of class field theory as a clue,exploring the infancy of the theory,sorting out the evolution of “class field”,analyzing the source of ideas,works and the impact of mathematicians.At the same time,the dissertation focuses on Furtw(?)ngler’s work and influence on reciprocity law,class field and the principal ideal theorem as a pioneer in the infancy of class field theory.The main results and conclusions are as follow:1.The beginning of class field theory is explored.Kronecker initiated the idea of“class field” and embodied it as a mathematical concept-the associated species.Later,Weber used the term “class field” for the first time to denote the associated species,and popularized its definition.In “Die Theorie der algebraischen Zahlk(?)rper”,Hilbert redefined the “class field”,and put forward a series of conjectures about the Abelian extensions on the number field in 1898.These conjectures were the primary form of the basic theorem of class field theory.2.Furtw(?)ngler’s life experience and his work in promoting the development of class field theory are examined.Furtw(?)ngler,a German mathematician,devoted himself to the study of number theory,especially achieved many important results on the Hilbert’s conjecture.He began to study Hilbert’s ninth problem in the early 20 th century,and generalized Hilbert’s quadratic reciprocity law to odd prime powers,primary ideals,odd prime exponents and even odd prime power exponents,thus partially solving Hilbert’s ninth problem.From 1907 to 1911,he proved the generality of Hilbert’s conjecture,that is,the existence of absolute class field,decomposition theorem and isomorphism theorem,which laid the foundation for Takagi Teiji’s establishment of class field theory.In 1930,based on Artin’s group theory,Furtw(?)ngler proved the principal ideal theorem,thus solving the problem left over for more than 30 years in class field theory.3.The process of Takagi Teiji’s establishment of class field theory is analyzed.Based on the research of Hilbert and Furtw(?)ngler,Takagi Teiji redefined the class field and established the connection between the abelian extension and the ideal class group of the base field,and put forward the basic theorem about class field theory,thus creating a new theoretical system-class field theory.4.The work by Artin,Hasse and Chevalley in simplifying and generalizing class field theory is discussed.Artin introduced Artin reciprocity law,which simplifies the proof of the basic theorem of class field theory.Hasse used the local-global principle to explore local class field theory.Chevalley introduced idèle’s concept,studied the topological properties of idèle,and fully arithmeticized class field theory with the help of idèle.5.The subsequent development of class field theory is introduced.As an important branch of algebraic number theory,class field theory has strong vitality.Initially,class field theory was only limited to solving the abelian extension problems on the number field.However,with the gradual improvement of the number theory system,mathematicians had studied deeply and found that the class field theory on the function field is also valid,and can even be extended to non-abelian extension.