Basic theorems in local class field theory and some further exploration

Research Article
Open access

Basic theorems in local class field theory and some further exploration

Peiwu Chen 1*
  • 1 Shandong University    
  • *corresponding author zjlscpw@sina.com
TNS Vol.5
ISSN (Print): 2753-8826
ISSN (Online): 2753-8818
ISBN (Print): 978-1-915371-53-9
ISBN (Online): 978-1-915371-54-6

Abstract

The relation between the abelian extension of a field and the topological groups of the field itself can be constructed using class field theory. In this piece of writing, the author will introduce the fundamental theorems of local class field theory by conducting a method of literature study. These theorems are the Reciprocity Law and the Existence Theorem, respectively. In addition to that, the author will discuss several unresolved issues in class field theory and provide examples of their applications in number theory. In class field theory, the results can be shown in two different ways. Both can be considered broad strokes. The first step is to demonstrate that the local case is true. By employing the methodology of cohomology and the theory of Lubin and Tate, one can demonstrate the Local Reciprocity Law and the Local Existence Theorem. The fundamental theorems in the global case are going to be demonstrated by utilizing the local results in conjunction with cohomology. Directly demonstrating the Gocal Reciprocity Law is another viable option.

Keywords:

local reciprocity law, local existence theorem, group cohomology, Lubin-Tate theory, local class field theory

Chen,P. (2023). Basic theorems in local class field theory and some further exploration. Theoretical and Natural Science,5,813-820.
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References

[1]. Milne, J.S. Class Field Theory. www.jmilne.org/math/. 2020:31-32,84.

[2]. Lubin, J. The Local Kronecker-Weber theorem. Trans. Amer. Math. Soc. 1981:133-138

[3]. Milne, J.S. Elliptic Curves. World Scientific Press, second edition. 2020

[4]. Weiss, E. Cohomology of groups. Pure and Applied Mathematics, Vol. 34. Academic Press, New York. 1969

[5]. Serre,J.-P. Local Class Field Theory. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C.1967:128-161.

[6]. Neukirch, J. Class Field Theory, volume 280 of Grundlehren der Mathematishen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin. 1986

[7]. Lang, S. Algebraic Number Theory. Addison-Weysley Publishing Co.,Inc., Reading, Mass.-London-Don Mills, Ont. 1970

[8]. Grant,K. And Leitzel,J. Nrom limitation theorem of class field theory. J. Reine Angew. Math. 238:105-111. 1969

Cite this article

Chen,P. (2023). Basic theorems in local class field theory and some further exploration. Theoretical and Natural Science,5,813-820.

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About volume

Volume title: Proceedings of the 2nd International Conference on Computing Innovation and Applied Physics (CONF-CIAP 2023)

ISBN：978-1-915371-53-9(Print) / 978-1-915371-54-6(Online)
Editor：Marwan Omar, Roman Bauer
Conference website: https://www.confciap.org/
Conference date: 25 March 2023
Series: Theoretical and Natural Science
Volume number: Vol.5
ISSN：2753-8818(Print) / 2753-8826(Online)

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References

[1]. Milne, J.S. Class Field Theory. www.jmilne.org/math/. 2020:31-32,84.

[2]. Lubin, J. The Local Kronecker-Weber theorem. Trans. Amer. Math. Soc. 1981:133-138

[3]. Milne, J.S. Elliptic Curves. World Scientific Press, second edition. 2020

[4]. Weiss, E. Cohomology of groups. Pure and Applied Mathematics, Vol. 34. Academic Press, New York. 1969

[5]. Serre,J.-P. Local Class Field Theory. In Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C.1967:128-161.

[6]. Neukirch, J. Class Field Theory, volume 280 of Grundlehren der Mathematishen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin. 1986

[7]. Lang, S. Algebraic Number Theory. Addison-Weysley Publishing Co.,Inc., Reading, Mass.-London-Don Mills, Ont. 1970

[8]. Grant,K. And Leitzel,J. Nrom limitation theorem of class field theory. J. Reine Angew. Math. 238:105-111. 1969

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