The Historical Evolution Of The Local-Global Principle

Algebraic number theory is one of the main branches of modern mathematics.The local-global principle is a powerful research method in algebraic number theory that links the properties of the local fields and the global fields.It not only greatly promotes the development of algebraic number theory,but also plays a pivotal role in modern mathematical theories such as algebraic arithmetic,arithmetic geometry,and Galois homology.Tracing the origin of the local-global principle can not only help people understand the motivation,ideological connotation,development,and evolution of the local-global principle,but also deeply understand the ideological inheritance and innovative spirit of mathematicians,making it helpful for people to master its applications in modern mathematics.Therefore,it has important theoretical value and practical significance to research the early history of the local-global principle.In history,for the origin,formation and development of the local-global principle,many mathematicians have made fruitful achievements one after another,among which the achievements of Kummer,Hensel,and Hasse are the most prominent.Based on a large number of relevant German original documents and research documents,this dissertation comprehensively uses research methods such as chronicles,conceptual analysis,charts,bibliographic statistics,and sociology to research the motivation,ideological connotations and development of the local-global principle from the 19th century to the early 20th century.The main research results and conclusions of this dissertation are as follows:1.This dissertation researches the ideological origin of the local-global principle.By analyzing Kummer's work on Fermat's Last Theorem in number theory,it is found that Kummer's proof methods for Fermat's Last Theorem involved the decision conditions of factorization of rational prime numbers in Z(?)and the definition of local uniformizing elements in Z(?),and creatively used the localized ideas.This was the ideological origin of the local-global principle.2.This dissertation analyzes the motivation and method of the p-adic number proposed by Hensel.By sorting and drawing a family map,this dissertation explores Hensel's excellent family background,sorts out the edification and positive influences of different characters on Hensel,and concludes that the influence of family members on Hensel was indispensable.By sorting out the relationships between Hensel's teachers at different stages of study,this dissertation finds that many of his teachers are reputable mathematicians in the mathematical community,who has a positive impact on the formation of his mathematical literacy and interest.In particular,Hensel devoted himself to academic research.By comparing the results of single variable algebraic number theory and algebraic function theory,he applied the power series expansion method in the function field to the number field,gave the p-adic expansion of algebraic numbers,and put forward the p-adic numbers,which provided a necessary premise for obtaining the local-global principle.3.This dissertation discusses the preliminary development of Hensel's theory of p-adic numbers.It is considered that Hensel gave a systematic theory of p-adic numbers through two works,Theorie der algebraischen Zahlen and Zahlentheorie,published in 1908 and 1913,respectively.The publication of Zahlentheorie attracted Hasse to conduct in-depth research on the theory of p-adic numbers.In addition,Fraenkel's axiomatization of p-adic number theory and Kürschák's theory of valuation theory also laid a solid theoretical foundation for p-adic numbers.4.This dissertation analyzes the process of Hasse's further proposing the local-global principle under the guidance and help of Hensel and focuses on exploring the change process of his mathematical thought.It is considered that Hasse was inspired by Hensel's reply to his own postcard and realized that there may be a connection between the nature of arithmetic in the field of p-adic numbers and the field of rational numbers.To this end,Hasse draws on Dedekind's problem-solving idea,using the Lagrange reduction principle and Hensel's criterion to obtain the local-global principle of quadratic form for the first time.On this basis,Hasse further developed the local-global principle in subsequent research.5.This dissertation expounds the practical applications of the local-global principle in algebraic arithmetic,arithmetic geometry,and Galois homology theory.In the proof of the main theorem of algebraic,the local-global principle plays a key role in simplification.In arithmetic geometry,the local-global principle facilitates research on quadratic forms and central simple algebras.In Galois homology,people combine the local-global principle on torsion,discrete valuations,and the Galois cohomology group with cohomology invariants to create new applications of the local-global principle.