Early Historical Research On The Principal Genus Theorem

The principal genus theorem is one of the most famous theorems in Gauss’ s work on genus and which is the central theorem of genus theory.This theorem originated from a conjecture in the communication between Euler and Goldbach,which was first proposed and proved by Gauss in his Disquisitiones Arithmeticae published in 1801.It was later extended by mathematicians like Dedekind,Kummer,Hilbert,Emmy Noether and Hasse,etc.The principal genus theorem is the algebraic basis for deriving the reciprocity law,and it is also an important cornerstone in the construction of the class field theory.Based on a large number of original literature and research literature,this dissertation uses research methods such as literature research method,chronicle method,sociological method and conceptual analysis method to explore the origin of the principal genus theorem with the early historical development of this theorem as the clue.Besides,this dissertation also combs the development and promotion process of this theorem,and analyzes the origin of thought and main work of mathematicians.At the same time,the application of the principal genus theorem is briefly introduced in this dissertation.The main results and conclusions are as follows:1.The origin of the principal genus theorem is explored.The study of using binary quadratic forms to represent integers originated from Fermat.In 1753,a conjecture about this content mentioned in the communication between Goldbach and Euler contained the idea of genus theory.In addition,Euler also proposed the concept of suitable numbers,and Gauss later gave two properties related to suitable numbers.Then,Lagrange introduced the concepts of equivalence and reduction in the theory of binary quadratic forms,which laid the foundation for the proposition of Gauss’ s genus theory.Besides,Gauss also proved the principal genus theorem in Disquisitiones Arithmeticae by using Legendre’s theory of ternary quadratic forms.2.The process of Gauss putting forward the principal genus theorem is analyzed.In the sixth chapter of Disquisitiones Arithmeticae,Gauss successively gave the concepts of character,genus and the composition of forms,discussed the quantity of ambiguous classes and obtained the first and the second inequalities of genus theory.Later,he deduced the principal genus theorem on binary quadratic forms and proved it.3.The works of Dedekind,Kummer and Hilbert on the development of the principal genus theorem are discussed.In 1839,Dirichlet gave an analytical proof of the second inequality of genus theory in “Recherches sur diverses applications de l’analyse infinitésimaleà la theorie des nombres”.With the influence of Dirichlet,Dedekind further developed Gauss’ s principal genus theorem for binary quadratic forms in the appendixes of Vorlesungenüber Zahlentheorie,and proposed an equivalent theorem for this theorem.Finally,he proved the principal genus theorem using Legendre theorem.Kummer studied the principal genus theorem on Kummer extension,and gave a general principal genus theorem on this extension.Hilbert defined the norm residue symbol and the division of ideal classes,thus deriving the principal genus theorem on the quadratic field.In addition,in the fifth part of Zahlbericht,he gave the principal genus theorem on regular Kummer field combined with Kummer’s results.4.The works of Hecke,Emmy Noether,Hasse and others on simplifying and extending the principal genus theorem are analyzed.Hecke defined the strict equivalence and gave a new concept of the genus,thus the principal genus theorem was simplified.By using crossed products,Emmy Noether gave the minimum principal genus theorem and the division of ideal classes induced by the factor system.On this basis,the principal genus theorem was extended to the relative Galois number field.Hasse generalized the principal genus theorem to the relative cyclic field of prime degree,and obtained the general principal genus theorem on this field.5.The application of the principal genus theorem is introduced.The principal genus theorem played a fundamental role in deriving the classical reciprocal law,and laid a foundation for Takagi and others to construct class field theory.