This document is a continuation of my Semester 1 project on class field theory. In the previous work, we made a rounded exposition of the fundamentals of class field theory but in order to preserve the document length the main proofs had to be skipped. We concentrate on filling in the gaps in this second installment. Due to the need to complete the arguments left open last semester and the need for applications this part of the project is a little longer than it should have been. It was not mentioned in the previous project but the class field theory we are studying here is global class field theory. There is such a thing as local class field theory in which we study the Abelian extensions of local fields (essentially fields that arise as completions of a number field with respect to places). Actually we touch on these ideas slightly in this project but never quite get to de_ning a local Artin map and looking at the local analogues of the main theorems of global class field theory. For those wanting to continue on to study local class field theory, consider Chapter 7 of [2] To start off this project we shall first restate the main de_nitions and theorems. This will be brief and those wanting to remind themselves of the details should consult my Semester 1 project. There will be very little motivation or technical results here since this was the purpose of the work done previously. We then set out to prove the main theorems of class field theory. With our present knowledge this would not be a simple task and we soon find that we first have to invent or discover new concepts such as the idele group and the corresponding idele class group. These are topological devices that take stock of all completions of a number eld at once. Such constructions will make the theory much easier to understand and formulate, whilst at the same time generalising the theory to all Abelian extensions. The cohomology of nite Abelian groups will be introduced and used alongside the idele theory to establish an important inequality. We use L-series in conjunction with the ideal theory to establish another important inequality. Combining the two inequalities will give a nice result that allows us to prove Artin reciprocity. In order to prove the existence theorem we resort to using Kummer n-extensions and the notion of a class eld. This middle chunk of the project will be quite technical but hopefully enjoyable and illuminating. [...]
Inhaltsverzeichnis (Table of Contents)
- Introduction
- Reminder of global class field theory
- Underlying theory
- The Artin map
- The main theorems
- The path to the idelic view
- Ideles
- Cohomology of finite cyclic groups
- Galois actions on ideles
- Proving the main results
- The universal norm index inequality
- The global cyclic norm index inequality
- Proving the Artin reciprocity law
- Proving the existence theorem.
- Primes of the form x² + ny²
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This document provides a comprehensive look at the proofs and applications of class field theory. It builds upon the previous work, filling in the gaps by providing detailed proofs of the main theorems. The focus shifts towards global class field theory, examining the Abelian extensions of global fields, with a brief touch on local class field theory. The project aims to explore the relationship between generalized ideal class groups and Abelian extensions, culminating in an application of the theory to solve a specific problem in number theory.
- The Artin reciprocity law and its role in establishing a correspondence between generalized ideal class groups and Abelian extensions.
- The concept of ideles and their application in simplifying and generalizing the theory of class field theory.
- The use of cohomology of finite Abelian groups and L-series in proving key inequalities leading to Artin reciprocity.
- The application of class field theory to determine which rational primes can be expressed in the form x² + ny².
- The significance of class field theory in other areas of mathematics, including the Cebotarev density theorem and higher reciprocity laws.
Zusammenfassung der Kapitel (Chapter Summaries)
- Introduction: This chapter introduces the project and its scope, providing context and highlighting the main goals of the work. It also briefly discusses the connection with previous work on class field theory and introduces the key concepts of global and local class field theory.
- Reminder of global class field theory: This chapter serves as a brief review of the fundamental concepts and theorems of global class field theory, including the definitions of moduli, congruence subgroups, generalized ideal class groups, and the Artin map.
- The path to the idelic view: This chapter introduces the concept of ideles, which provide a more concise and general framework for studying class field theory. It defines ideles, discusses their properties, and outlines their relevance to understanding Abelian extensions.
- Proving the main results: This chapter focuses on the proofs of the two main theorems of class field theory, the Artin reciprocity theorem and the existence theorem. The proof strategy involves deriving key inequalities using L-series and cohomology, and ultimately utilizing Kummer n-extensions to prove the existence theorem.
- Primes of the form x² + ny²: This chapter presents an application of class field theory to solve a specific number theory problem. It demonstrates how the theory can be used to determine which rational primes can be expressed in the form x² + ny² for specific values of n.
Schlüsselwörter (Keywords)
The core concepts and focus areas of this project revolve around class field theory, Abelian extensions, Artin reciprocity law, ideles, cohomology, L-series, Kummer n-extensions, generalized ideal class groups, and the application of class field theory to number theory problems such as determining primes expressible in the form x² + ny².
- Quote paper
- Daniel Fretwell (Author), 2011, Class Field Theory: Proofs and Applications, Munich, GRIN Verlag, https://www.grin.com/document/175761
