The Early Historical Research On Valuation Theory

Valuation theory is one of the important branches of algebraic number theory and a tool to study commutative algebra.It has very important applications in class field theory,ramification theory and algebraic geometry.It originated in Kürschák and was developed through the work of Ostrowski and others.Later,Krull established general valuation theory,which laid the foundation of valuation theory.Therefore,studying the early history of valuation theory will help us clarify the development of valuation theory and understand its role in the development of other mathematical theories.On the basis of a large amount of original literature and research literature,this dissertation uses the methods of concept analysis,chronicle,literature research,chart and sociology,and takes the ideological history of valuation theory as the clue.This dissertation explores the origin and development of valuation theory and the establishment of general valuation theory in detail from the evolution of the concept of valuations,the early thoughts of valuations of mathematicians and the mutual promotion of the development and applications of valuation theory.Furthermore,the applications of valuation theory are also briefly introduced.The main results and conclusions are as follows:1.The origin of valuation theory is explored.In 1912,inspired by Henzel's Algebraic Number Theory,Hungarian mathematician Kürschák made a report entitled“über Limesbildung und allgemeine K?rpertheorie” at the International Congress of Mathematicians in Cambridge,and put forward the concepts of “valuation” and“valued field” for the first time.Since then,valuation theory had appeared on the algebraic stage as an independent branch of mathematics.Later,many mathematicians began to study valuation theory under the influence of Kürschák.2.The work of Ostrowski and Rychlík in the development of valuation theory is explored.Ostrowski first solved the problem left by Kürschák's paper,then classified the concept of valuations,put forward the basic theorem of valuation theory,and further developed the theory.The Czech mathematician Rychlík generalized their results and restated some of Henzel's results in the framework of an arbitrary complete field with a non-archimedean valuation.Many years later,Ostrowski studied valuation theory again and got some new results,which were collated and published together with results which were obtained in his previous research but not publicly.Many concepts and theories in the paper were also extremely important in general valuation theory.3.The work of Hasse and Schmidt in the development of valuation theory is analyzed.After some development of valuation theory,it had been widely used in number theory.Hasse discovered the local class field theory by using valuation theory.In order to obtain further results,he also studied the structure of complete valued fields with Schmidt,and obtained the structure theorem of discrete valued complete fields.Later,Mac Lean studied the proofs of Hasse and Schmidt,gave a new proof of the structure theorem,and proposed the important concept of “separable field extension”.4.The establishment process of general valuation theory is explored.In 1926,Artin and Schreier proved the existence of real fields.At that time,Krull realized that the concept of valuations defined by Kürschák could not describe all the characteristics of real fields,so he generalized the definition of valuations,proposed the concept of general valuations,and established general valuation theory.Since then,he continued to study the theory,and affected his students and other mathematicians to study valuation theory,which laid a foundation for the wide spread and applications of valuation theory.5.The applications of valuation theory are analyzed.The earlier applications of valuation theory were in number theory and abstract algebra.Hasse applied valuation theory to number theory and got the local-global principle and local class field theory.Krull also applied valuation theory to get some conclusions about integrally closed rings and real fields.Later,the applications of valuation theory in algebraic geometry and model theory were becoming mature.Zariski and his student Abhyankar made important progress in the reduction of singularities by using the idea of valuations,and the use of valuation methods for p-adic fields by Ax,Kochen,Ershov and Macintyre laid a foundation for the applications of valuation theory in model theory.