Algebra in Action: A Course in Groups, Rings, and Fields

This textbook was designed for an undergraduate two-semester course sequence for students learning modern algebra for the first time. The tone of the writing is conversational but nevertheless also includes the traditional style of presenting concepts using definitions, the theorem, a proof, and subsequently examples. The author incorporated three defining features in his book: group actions are introduced early (Chapter 4), Hasse diagrams of posets are extensively used, and normal subgroups, quotients groups, and homomorphisms are introduced later. This textbook approaches algebra in a speedy and sophisticated manner, and introduces advanced concepts and a great number of new notations. It may potentially be overwhelming for inexperienced undergraduates. This style is quite different from conventional textbooks that focus on the most basic and essential elements of the subject.

The first chapters cover the history and reason for studying modern algebra and how it grew from different areas of study. This is followed by introducing groups, cyclic groups, isomorphisms, subgroups, and other critical concepts. These introductory chapters are meant to be read by students; the author suggests that instructors quickly finish these parts to proceed to the next chapters. Unlike in other introductory textbooks, these topics are not thoroughly explained with examples and pictures. Problems were included to be solved by the students on their own. It seems that the author wanted to move very quickly to more advanced topics. For those deciding to use this textbook in class, it should be noted that it requires that students have a strong mathematical background on reading and writing proofs, and perhaps to have taken other upper-level mathematics courses. They certainly need to be strong and academically independent.

This textbook has plenty of challenging problems for students to work on. The author encourages self-study and provides hints and short answers; for some problems complete solutions are provided. Working on the problems will leave the students with a deep understanding of the subject matter and mathematical proficiency that will carry over to graduate level courses. The author designed this book to prepare mathematically strong undergraduate students to pursue further graduate level courses and perhaps a higher degree in mathematics.

In conclusion, instructors should choose this textbook if they are teaching a two-semester algebra sequence for students who are well prepared to tackle an introductory topic in a more advanced manner. This textbook might not be suitable for students who need more time to review other mathematical topics such as concepts necessary to understand group theory, and reading and writing proofs. It may also not be appropriate for students who need a slower pace and more extensive explanation for critical concepts. It is rigorous, well-written, ample in terms of problems and solutions provided, and sufficiently advanced for its target audience.

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Mark Causapin is an Assistant Professor of Mathematics at Aquinas College in Nashville, Tennessee.