Abstract Algebra

Introduction to Groups, Rings and Fields with Applications

This second edition covers essentially the same topics as the first. However, the presentation of the material has been extensively revised and improved. In addition, there are two new chapters, one dealing with the fundamental theorem of finitely generated abelian groups and the other a brief introduction to semigroup theory and automata.

This book is appropriate for second to fourth year undergraduates. In addition to the material traditionally taught at this level, the book contains several applications: Polya–Burnside Enumeration, Mutually Orthogonal Latin Squares, Error-Correcting Codes, and a classification of the finite groups of isometries of the plane and the finite rotation groups in Euclidean 3-space, semigroups and automata. It is hoped that these applications will help the reader achieve a better grasp of the rather abstract ideas presented and convince him/her that pure mathematics, in addition to having an austere beauty of its own, can be applied to solving practical problems.

Considerable emphasis is placed on the algebraic system consisting of the congruence classes mod n under the usual operations of addition and multiplication. The reader is thus introduced — via congruence classes — to the idea of cosets and factor groups. This enables the transition to cosets and factor objects to be relatively painless.

In this book, cosets, factor objects and homomorphisms are introduced early on so that the reader has at his/her disposal the tools required to give elegant proofs of the fundamental theorems. Moreover, homomorphisms play such a prominent role in algebra that they are used in this text wherever possible.

Contents:

• Logic and Proofs
• Set Theory
• Cartesian Products, Relations, Maps and Binary Operations
• The Integers
• Groups
• Further Properties of Groups
• The Symmetric Groups
• Rings, Integral Domains and Fields
• Polynomial Rings
• Field Extensions
• Latin Squares and Magic Squares
• Group Actions, the Class Equation, and the Sylow Theorems
• Finitely Generated Abelian Groups
• Semigroups and Automata
• Isometries
• Pólya-Burnside Enumeration
• Group Codes
• Polynomial Codes
• Appendix A: Rational, Real, and Complex Numbers
• Appendix A: Linear Algebra

Readership: Undergraduates from approximately 2nd to 4th year. Familarity with linear algebra is required.
Key Features:

• Emphasis on congruences in the integers with the view of introducing cosets and factor objects early on in the book
• Several applications to demonstrate the power of the abstract ideas introduced
• Inclusion of many challenging problems, often with hints of how to solve them. Many introductory texts give a plethora of routine problems but very few interesting harder ones
• The unifying theme of homomorphisms which are introduced early on in the text after spending a considerable amount of time computing and proving theorems in the integers mod n
• Inclusion of two appendices to make the book as self-contained as reasonably possible. The book assumes that the reader is familiar with the rudiments of the rational, real and complex numbers and linear algebra. Appendix A gives a quick and "naïve" account of some of the relevant facts about the familiar number systems. In Appendix B, the reader will find a fairly extensive treatment of linear algebra, emphasizing linear transformations and their relationship to their matrix representations