This is the first in a two part series of papers establishing (with proof) the main theorems of global class field theory. We first recap some of the main ideas of algebraic number theory, using these to develop the Artin reciprocity law and the Takagi existence theorem both in terms of ideals and ideles. Finally, we use the Hilbert class field in order to study the well known problem of which prime numbers can be represented in the form x^2 + ny^2 for integers x,y and positive integer n.
Table of Contents
1 Introduction
2 A brief survey of algebraic number theory
3 The ideal class group
3.1 Fractional ideals
3.2 The ideal class group
4 The decomposition group and Frobenius
4.1 Galois theory of finite fields
4.2 A Galois group action
4.3 The Artin symbol
5 Class field theory
5.1 Class field theory in unramified abelian extensions
5.2 Loosening the unramified condition
Objectives and Research Themes
This work aims to provide an accessible introduction to Class Field Theory, focusing on the interplay between algebraic number fields and their Abelian extensions. The primary research question explores how the arithmetic of a number field—specifically its ideal class groups—can classify and describe all Abelian extensions of that field.
- Foundations of algebraic number theory and the role of the ideal class group.
- Galois group actions and the study of decomposition groups and Frobenius automorphisms.
- The definition and application of the Artin symbol and the Artin map.
- The classification of unramified Abelian extensions and the existence of the Hilbert class field.
- Generalization of the theory through the introduction of moduli and the Artin reciprocity theorem.
Excerpt from the Book
3.2 The ideal class group
Now that we have defined the fractional ideals of a number field K, we are ready to construct the ideal class group of K. As mentioned, this group measures the extent to which unique factorisation of elements fails. The basic idea follows from the fact that unique factorisation into irreducibles holds if and only if every ideal is principal. So if we can somehow measure how far a given ideal is from being principal then we can use this information to see how much unique factorisation fails.
The best way forward is to construct the quotient group IK/PK, where PK is the group of non-zero principal fractional ideals (i.e. fractional ideals of the form (α), where α ∈ K×).
Definition 3.2.1. The group IK/PK is called the ideal class group of K, denoted ClK.
Surprisingly enough we have the following, non-trivial result:
Theorem 3.2.2. The group ClK is a finite group.
Summary of Chapters
1 Introduction: Provides a brief overview of the historical goal to classify finite extensions of number fields and introduces the core concept of class field theory.
2 A brief survey of algebraic number theory: Reviews essential definitions such as number fields, rings of integers, and the unique factorization of ideals into prime ideals.
3 The ideal class group: Defines fractional ideals and constructs the ideal class group to measure the failure of unique factorization.
4 The decomposition group and Frobenius: Examines prime ideal factorization in Galois extensions and introduces the decomposition group and Frobenius automorphisms.
5 Class field theory: Details the Artin map, the Hilbert class field, and generalizes the theory using moduli to study ramified Abelian extensions.
Keywords
Class Field Theory, Number Fields, Ideal Class Group, Galois Theory, Abelian Extensions, Artin Map, Artin Symbol, Frobenius Automorphism, Hilbert Class Field, Modulus, Ray Class Field, Conductor, Artin Reciprocity, Ramification, Ring of Integers
Frequently Asked Questions
What is the fundamental focus of this work?
This work focuses on Class Field Theory, which seeks to describe and classify all finite Abelian extensions of a given number field in terms of the field's internal arithmetic.
What are the core thematic fields covered?
The core themes include algebraic number theory, Galois theory, the structure of ideal class groups, and the behavior of prime ideal ramification in extensions.
What is the primary objective of this research?
The primary objective is to explain how the Artin map establishes a bridge between the group of fractional ideals (or its generalized versions) and the Galois groups of Abelian extensions.
Which scientific methods are primarily employed?
The work utilizes methods from algebraic number theory and group theory, specifically examining quotient groups of fractional ideals and the properties of automorphisms acting on finite field extensions.
What is discussed in the main part of the text?
The main part covers the construction of the ideal class group, the transition to decomposition groups and Frobenius elements, and the formalization of the Artin map to classify both unramified and ramified Abelian extensions.
Which keywords best characterize this publication?
Key terms include Class Field Theory, Artin map, Hilbert class field, ray class field, and Abelian extensions.
What role does the Artin symbol play in this theory?
The Artin symbol serves as a generator for the decomposition group in unramified extensions, allowing for a precise description of how prime ideals factorize in Galois extensions.
Why is the introduction of a "modulus" necessary?
The modulus is a formal tool used to track ramification; it allows the theory to be generalized from unramified Abelian extensions to include ramified cases by "filtering" information through specific congruences.
- Quote paper
- Daniel Fretwell (Author), 2011, Class Field Theory, Munich, GRIN Verlag, https://www.grin.com/document/175757
