Class Field Theory: Proofs and Applications

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.