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Introduction to Algebra

Peter J. Cameron
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Miklós Bóna
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This book is a pleasure to read. Its chapters are by and large self-contained, the examples are well-chosen and not overused, and the style is reader-friendly. The exercises are interesting, too.

For classroom use goes, however, there are a few problems because the author chose a path that is very unusual for an Abstract Algebra textbook meant for universities in the US. The first chapter, Introduction, covers proof techniques and logic. Most students taking this class would already have seen these, in a Transition to Advanced Mathematics course, which is usually a prerequisite for this course. This could still be helped by simply skipping that chapter. Then the author discusses Rings, one chapter earlier than he discusses Groups. This reviewer would prefer a book that covers these two topics in the reverse order, since groups are not as strongly structured as rings. Fields are relegated to one section within Rings, which is another choice that many would question.

After these important chapters, there is a chapter on vector spaces. Most universities offer, or even require, a course in Linear Algebra, in which vector spaces are covered at great length. Therefore, many instructors will be inclined to skip this chapter as well. That may not leave enough material for a full (14-week) semester.

The book ends by chapters on Modules, Number Systems, Applications to Coding Theory and Galois Theory, and by mentioning further directions in group theory, ring theory, and field theory. The difficult proof of the fact that e is not an algebraic is very well explained.

To summarize, this reviewer believes that the vast majority of readers will enjoy this book, even if most of them will not adopt it for classroom use.

Miklós Bóna is Associate Professor of Mathematics at the University of Florida.

1. Introduction
2. Rings
3. Groups
4. Vector spaces
5. Modules
6. The number systems
7. Further topics
8. Applications
Further reading