LECTURE NOTES OF WILLIAM CHEN

 

INTRODUCTION TO ALGEBRAIC NUMBERS

This set of notes has been used between 1981 and 1990 by the author at Imperial College, University of London.

The material has been organized in such a way to create a single volume suitable for an introduction to the mathematics surrounding early attempts to understand Fermat's last theorem that lead to the development of ideal theory.

To read the notes, click the links below for connection to the appropriate PDF files.

The material is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, the documents may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners.

Chapter 1 : POLYNOMIALS >>

• Divisibility
• Irreducubility
• Polynomials with Rational Coefficients
• Symmetric Polynomials

Chapter 2 : ALGEBRAIC NUMBERS AND ALGEBRAIC INTEGERS >>

• Gaussian Integers
• Field Extensions
• Algebraic Numbers
• Conjugates
• Algebraic Integers
• Discriminants and Integral Bases
• Quadratic Number Fields
• Cyclotomic Number Fields
• Factorization

Chapter 3 : IDEAL THEORY >>

• Introduction
• Ideals in an Algebraic Number Field
• The Classical Proof of the Unique Factorization Theorem
• The Modern Proof of the Unique Factorization Theorem
• Consequences of the Unique Factorization Theorem
• Norm of an Ideal

Chapter 4 : CLASS GROUP AND CLASS NUMBER >>

• Fractional Ideals
• Some Geometric Input
• Ideal Classes
• Consequences of the Finiteness of the Class Number

Chapter 5 : KUMMER'S THEOREM >>

• Ramification
• Units in Cyclotomic Fields
• A Special Case of Fermat's Last Theorem