Class Field Theory: Proofs and Applications

Table of Contents

1 Introduction

2 Reminder of global class field theory

2.1 Underlying theory

2.2 The Artin map

2.3 The main theorems

3 The path to the idelic view

3.1 Ideles

3.2 Cohomology of finite cyclic groups

3.3 Galois actions on ideles

4 Proving the main results

4.1 The universal norm index inequality

4.2 The global cyclic norm index inequality

4.3 Proving the Artin reciprocity law

4.4 Proving the existence theorem

5 Primes of the form x^2 + ny^2

5.1 A theoretical solution to the problem

5.2 Three examples

Research Objectives and Core Themes

This work aims to provide detailed proofs for the fundamental theorems of global class field theory, building upon a preceding exposition of the theory's foundations. The central research question focuses on establishing the Artin reciprocity law and the existence theorem through the study of ideles and cohomology, eventually applying these results to characterize rational primes that can be represented by specific quadratic forms.

• The construction and utilization of ideles and idele class groups for global class field theory.
• Applications of the cohomology of finite cyclic groups to derive norm index inequalities.
• Proof of the Artin reciprocity law and the existence theorem as the two main pillars of the correspondence.
• Characterization of rational primes of the form x² + ny² using Hilbert class fields.

Excerpt from the Book

Summary of Chapters

1 Introduction: Provides context for this work as a continuation of previous studies on class field theory, focusing on filling in technical proofs.

2 Reminder of global class field theory: Recapitulates fundamental definitions, the Artin map, and the main theorems that form the starting point of the current investigation.

3 The path to the idelic view: Introduces ideles as essential topological tools for simplifying and generalizing class field theory, alongside relevant cohomological concepts.

4 Proving the main results: Develops the formal proofs for the norm index inequalities, Artin reciprocity, and the existence theorem using idelic and cohomological methods.

5 Primes of the form x^2 + ny^2: Demonstrates an application of the established theory to solve representation problems for primes using Hilbert class fields.

Keywords

Class Field Theory, Ideles, Artin Reciprocity, Existence Theorem, Galois Group, Cohomology, Norm Index Inequality, Hilbert Class Field, Dedekind Zeta Function, L-series, Primes, Quadratic Forms, Modulus, Abelian Extensions, Number Fields

Frequently Asked Questions

What is the primary focus of this document?

The work focuses on completing the formal proofs for the foundational theorems of global class field theory, specifically the Artin reciprocity law and the existence theorem, which were skipped in preceding work.

What are the central themes discussed?

The core themes include the idelic formulation of class field theory, the cohomology of cyclic groups, norm index inequalities, and the arithmetic application of Hilbert class fields to prime number representation.

What is the central research question?

The project seeks to establish the complete correspondence between Abelian extensions of number fields and generalized ideal class groups, utilizing ideles and cohomology as the primary technical machinery.

What scientific methods are employed?

The author uses idelic analysis, cohomology of finite cyclic groups, the study of L-series, and the properties of Hilbert class fields to rigorously derive the necessary norm inequalities.

What topics does the main body cover?

The main body systematically develops the idelic view, proves the Artin reciprocity theorem, establishes the existence theorem, and provides an application involving primes of the form x² + ny².

Which keywords best characterize this work?

Key terms include Class Field Theory, Ideles, Artin Reciprocity, Galois Group, Hilbert Class Field, and Number Fields.

How is the transition from ideal theory to idelic theory justified?

The transition is justified by the fact that the idelic view provides a more concise formulation of the theory and allows for the generalization to infinite extensions, which is more difficult with traditional ideal theory.

What role does the Hilbert class field play in the applications?

The Hilbert class field acts as a maximal unramified Abelian extension, which allows for the classification of primes representable by the quadratic form x² + ny² based on their splitting behavior in this extension.