The Early Historical Research On Dirichlet’s Unit Theorem

Algebraic number theory is one of the important branches of mathematics,and Dirichlet’s unit theorem is a fundamental theorem in algebraic number theory,which describes the basic structure of the group of units in an arbitrary algebraic number field.This theorem occupies a pivotal position in the history of mathematics and the proposal of it has greatly promoted the development of algebraic number theory.Tracing it back to the source of Dirichlet’s unit theorem can not only help people understand its genesis,ideological connotation and the evolution process of thought,but also help people understand Dirichlet’s unit theorem in various forms of expression and grasp the essence of its mathematical thought.Furthermore,the research on this subject is also beneficial in helping people understand the application of the theorem to modern mathematics.Therefore,it is of great theoretical value and practical significance to research the early history of Dirichlet’s unit theorem.Historically,in the process of proposing and developing of Dirichlet’s unit theorem,many mathematicians participated in it and made great achievements,among which,Dirichlet was the proposer of the theorem,and mathematicians such as Bachmann,Dedekind,Kronecker,Hilbert,van der Waerden,Hasse and Chevalley made important contributions to its development.Based on a large number of primary documents and research literature in German,French and Latin,taking Dirichlet’s related work as the starting point of the research,this dissertation focuses on the birth and development of Dirichlet’s unit theorem in the 19 th century,through the comprehensive use of literature research method,chronicle method,conceptual analysis method,sociological method,comparative research method,chart method and other research methods.The main research results and conclusions of this dissertation are as follows:1.This dissertation researches the mathematical background leading to the birth of Dirichlet’s unit theorem.By analyzing the development process of mathematics at that time,it is found that the proposal of complex integers prompted Dirichlet to place what is now known as the algebraic integer at the center of his research,and to publish a series of related papers,which paved the way for his later formulation of the unit theorem.2.Dirichlet’s life experiences,mathematical achievements,character and scientific research habits are analyzed in this dissertation.It is found that Dirichlet was modest,rigorous and good at finding profound mathematical thoughts from simple things.Moreover,Dirichlet has been highly enthusiastic about mathematics since he was a teenager,seizing all the time to learn and study mathematical problems.His love for mathematics and his good habit of using all his time to think about mathematical problems efficiently were important factors for him to get the unit theorem and its general proof.3.This dissertation analyzes Dirichlet’s original papers on the unit theorem and finds that Dirichlet’s work on the unit theorem originated from his study on the prime number theorem of arithmetic progression.In the process of generalizing the prime number theorem of arithmetic progression,Dirichlet obtained the first result on the unit theorem,that is,if αis an algebraic integer with at least one real conjugate,then Z[α] has infinitely many units,which was the beginning of his study on the unit theorem.Later,in the process of further research and proof of this result,Dirichlet published several related papers,and finally proposed the unit theorem formally and gave a general proof in his paper on complex units published in 1846.In addition,this dissertation also briefly describes the work of other mathematicians on the unit theorem who lived at the same time as Dirichlet,such as Eisenstein,Kronecker and Hermite.4.This dissertation explores the related work of mathematicians such as Bachmann,Dedekind,Kronecker,Hilbert,et al.,and analyzes the improvements they made in theorem expression and theorem proof.It is found that compared with Dirichlet’s original research results,the proof process of the unit theorem given by Bachmann et al.is more detailed and perfect.At the same time,the expression of the theorem also experienced the evolution from the relation of the roots of the algebraic equation to the relation of the unit,gradually approaching the modern expression of Dirichlet’s unit theorem which took groups as the object of expression,that is,the group of units of any algebraic number field is the direct product of its group of roots of unity and a free abelian group whose rank is finite.In addition,Hilbert not only gave the proof and a new expression of the theorem,but also made an important generalization of it.5.This dissertation expounds some further related research cases on the unit theorem.It is found that Minkowski and Herbrand got the unit theorem on Galois field,through the related research on the unit.In addition,mathematicians such as Hasse,Chevalley and Takashi Ono not only connected Dirichlet’s unit theorem with S units,solvable groups,algebraic tori,etc.,and formed corresponding generalization theorems,but also combined it with arithmetic varieties,algebraic dynamic systems and valuation theory.These works expanded the application range of Dirichlet’s unit theorem unceasingly,and also made the influence of it more far-reaching.