Groups, Rings and Galois Theory

✍ Scribed by Victor P Snaith

• Publisher: World Scientific Publishing Company
• Year: 2003
• Tongue: English
• Leaves: 228
• Edition: 2nd Revised ed.
• Category: Library

✦ Synopsis

This book is ideally suited for a two-term undergraduate algebra course culminating in a discussion on Galois theory. It provides an introduction to group theory and ring theory en route. In addition, there is a chapter on groups -- including applications to error-correcting codes and to solving Rubik's cube. The concise style of the book will facilitate student-instructor discussion, as will the selection of exercises with various levels of difficulty. For the second edition, two chapters on modules over principal ideal domains and Dedekind domains have been added, which are suitable for an advanced undergraduate reading course or a first-year graduate course.

✦ Table of Contents

Preface
Preface to the Second Edition
Contents
Chapter 1. Group Theory
1.1 The concept of a group
1.2 Exercises
1.3 New groups from old
1.4 Exercises
1.5 Quotient groups and normal subgroups
1.6 Exercises
1.7 Finitely generated abelian groups
1.8 Exercises
1.9 Abelian groups and codes
1.10 Exercises
1.11 Sylow's theorems
1.12 Exercises
1.13 Groups of permutations
1.14 Exercises
Chapter 2. Ring Theory
2.1 Basic definitions
2.2 Exercises
2.3 Factorisation
2.4 Exercises
2.5 Unique factorisation
2.6 Exercises
Chapter 3. Galois Theory
3.1 Fields
3.2 Exercises
3.3 Group characters
3.4 Exercises
3.5 Finite fields
3.6 Exercises
3.7 Further results
3.8 Exercises
3.9 Solution of equations by radicals
3.10 Exercises
3.11 Tensor products
3.12 Exercises
Chapter 4. Rings and Modules
4.1 Basic definitions
4.2 Finitely generated modules
4.3 Tensor products over rings
4.4 Exercises
Chapter 5. Dedekind Domains
5.1 Integrality
5.2 Prime factorisation of ideals
5.3 Finitely generated modules
5.4 Galois extensions
5.5 Exercises
Bibliography
Index