Elementary Abstract Algebra: Examples and Applications

Reviewed by Mark Koester, Associate Professor, Metropolitan State University of Denver on 3/26/21, updated 5/26/21

Comprehensiveness rating: 4 see less

Extensive index is included. The book includes more topics than a semester course could study.

Content Accuracy rating: 4

There appears to be accuracy in the book.

Relevance/Longevity rating: 5

These topics in Abstract Algebra are appropriate for secondary preservice mathematics teachers. Many of our state's mathematics standards connect to the topics in the book.

Clarity rating: 5

I enjoy the conversational style of the authors. It is so important that textbooks are accessible for all.

Consistency rating: 4

The same terminology is used throughout.

Modularity rating: 4

There are twenty chapters. Of course, some ideas build on others, but often, it is noted in what chapter these other ideas can be found.

Organization/Structure/Flow rating: 4

There is a logical order to the text.

Interface rating: 4

I like having the ability to click on hints or propositions n the book rather than being inundated with text on a page.

Grammatical Errors rating: 4

I didn't notice grammatical errors.

Cultural Relevance rating: 4

I believe that this book is inclusive of others because it is so readable. There is not the inherent bias that one must be able to read cryptic symbols or to read the author's words obviously, clearly, or this is easy. This was so damaging to having a positive identity as a mathematics major.

Reviewed by Diana Morris, Assistant Professor, University of Virginia on 12/13/20

Comprehensiveness rating: 5 see less

At almost 1000 pages, this book certainly covers the basics comprehensively. You might not find the Sylow Theorems or Jordan Canonical Form, but it accomplishes what it sets out to do, covering the basic background of numbers and set theory, the main examples of groups and rings, and then developing these concepts more formally.

Content Accuracy rating: 4

Given the trade-off between perfect mathematical clarity (think Rudin) and being intelligible to undergraduates, I think the book strikes a good balance.

Relevance/Longevity rating: 5

Abstract algebra books typically struggle to convince undergraduates of their relevance. The sustained application in the book is Cryptography, but they did a beautiful job of relating basic concepts like modular arithmetic to practical applications that we do every day (e.g. if December 1st is on a Tuesday, what day of the week is Christmas?)

Clarity rating: 5

Their basic wording of mathematical concepts was impressive, e.g. the explanation of multiplicative and additive identities is completely fool-proof, as were the every day examples of mathematical concepts, e.g. the notion of "grandfather" being a composition of "parent" and "father". There is a real effort to guide students through proofs, e.g. how to justify each step and what qualifies as justification. I also appreciated the generous hints on the exercises spread throughout the book and the Study Guides at the end of each chapter identifying competencies.

Consistency rating: 4

The material has numerous different sources. None of the transitions were jarring, but sometimes it felt more like a compendium than one flowing volume.

Modularity rating: 4

Given the length of the text, an instructor would certainly want to pull out different chapters and make particular note of which leads to which. There is actually a chart at the beginning of the text explaining how sections are related and which rely on previous sections.

Organization/Structure/Flow rating: 4

I could not decide if the organization of the book was confusing or brilliant. For example, major examples of groups / rings are first defined in great depth without formally defining them as such, then in later chapters, one starts from the beginning with the definitions of groups and rings and discovers that they are already familiar.

I did feel that there was an overwhelming amount of material, more a reference book of basic abstract algebra topics than a simple narrative, but there were highly coherent sub-narratives throughout.

Interface rating: 5

The only thing that made it difficult to navigate was the sheer length of the document.

Grammatical Errors rating: 4

The book had its share of misspellings, but not to the point of compromising or distracting from the meaning.

Cultural Relevance rating: 5

The book did an excellent job of relating abstract mathematical concepts to practical examples familiar to any person.

Comments

I would certainly recommend this book as a reference to any student wanting simple explanations and practical examples for the concepts of basic abstract algebra. For most single semester introductory courses, it would make a fine basic textbook, with some selective pruning and perhaps a few additions where the instructor sees fit.