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.

## 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

- Quote paper
- Daniel Fretwell (Author), 2011, Class Field Theory, Munich, GRIN Verlag, https://www.hausarbeiten.de/document/175757