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Algebra: Groups, Rings, and Fields

Louis Rowen
A K Peters
Publication Date: 
Number of Pages: 
[Reviewed by
William Porter
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Louis Rowen’s text, Algebra: Groups, Rings, and Fields, is written for undergraduates and covers the traditional two-semester sequence of courses in its namesake subject. Choosing to begin the body of the exposition by defining groups, Rowen uses this structure to establish the ideas central to the study of algebra, such as homomorphism and normality. After treating the classification of finite Abelian groups, the author spends a few more chapters discussing other important ideas in group theory before moving on to ring and field theory.

Though my opinion improved significantly throughout my reading, my initial impression of this text was somewhat less than favorable. The section preceding the definition of groups, entitled “prerequisites,” was more confusing than helpful, and it left me feeling as though I were stumbling into the bulk of the material. It presented so many properties and ideas so quickly that the constant referencing of equations made it almost impossible for me to read through, even after having had a year’s course in the subject.

My judgment of Rowen’s text became more and more favorable the more I read. His decision to prove Cauchy’s theorem early using a combinatorial argument provides access to this powerful tool early in the work’s development and offers the reader an example of this useful proof technique. Another choice of the author’s that I found particularly delightful was his integration of Noether’s isomorphism theorems and the Sylow theorems directly into the exposition rather then separating them off into appendices or footnotes.

The discussion that Rowen includes when he introduces the ideas of homo- and isomorphism and the one that precedes the chapter on Cayley’s theorem are examples of particularly insightful bits of writing. Throughout the book, the author takes a break from the proof-based style characteristic of mathematics texts and discusses the implications of one result or the problems with another.

Still, I’m not sure that I’d choose this text for an introductory course in algebra. At times, I felt that a lot of the proofs lacked sufficient explanation. Other sections, such as the one covering permutations, include few or no proofs. Had I not already covered the notation and the concepts presented there, I’m not sure I’d have felt comfortable with that material.

I’m not sure this text would be easy to use for someone who doesn’t already have a good grasp of the fundamental ideas. Cayley tables enjoy minimal use, and the tree-like diagrams usually used to explain subgroups, field extensions, and the ideas of Galois theory are for the most part absent.

There are myriad exercises in this book. They are as varied in subject as I’ve seen, and vary appropriately in difficulty. Further, several important ideas and theorems are covered using two or three exercises when they cannot be included in the text itself, the Chinese remainder theorem being an example. This is a great alternative to omitting them completely.

This book is an excellent, detailed treatment of abstract algebra. It contains all of the material traditionally included in the undergraduate course on groups, rings, and fields as well as application to many areas such as number theory and the problems of antiquity. Though it contains the necessary material, is well written, and is equipped with many wonderful exercises, I worry that the style, pacing, and lack of detail in crucial places could make this text difficult to learn from in an introductory course.

William Porter is an undergraduate mathematics-physics double major at a small liberal arts college. He enjoys ballroom dancing, cooking, and T. S. Eliot’s poetry. He thinks that Rachmaninoff writes amazing music and rain is the best weather. His favorite Big Bang Theory character is Sheldon.

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