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. [...]
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
4.1 The universal norm index inequality
In this subsection we prove the first of the two inequalities that will be useful later. In order to tackle the proof we require specific L-series constructed from the characters of the finite group IK(m)/PK,1(m)NL/K(IL(m)) (where L/K is an Abelian extension of number fields and m is a complete modulus for L/K).
No preliminary knowledge of L-series is needed although I shall only motivate the results here. The material is not directly important to this project but an interested reader can consult Chapter VIII of [1] for proofs and discussions that are omitted.
Recall that given a sequence {an} of complex numbers we can define the corresponding Dirichlet series: L(s) = Σ an/n^s, where s is a complex variable.
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.
- Citar trabajo
- Daniel Fretwell (Autor), 2011, Class Field Theory: Proofs and Applications, Múnich, GRIN Verlag, https://www.grin.com/document/175761
