A Course on Abstract Algebra

In the preface to the text, the authors state that their primary motivation in writing a textbook in abstract algebra was to aid their students for whom English is a second language. They set themselves the worthwhile goal of presenting the major topics of abstract algebra in as unadorned fashion as possible and providing thoughtful and well-written exercises.

All the essential topics in abstract algebra are covered in a traditional fashion. The authors begin with an extensive treatment of groups, before presenting material on rings, vector spaces, and fields. The text is designed for a fast-paced one year course or a three semester introductory sequence in abstract algebra.

The authors present clear explanations and proofs of the major concepts and results in abstract algebra. They begin with a chapter of preliminary material, including the Well-Ordering Principle and the Principle of Mathematical Induction (without proofs). They include a chapter on free groups and presentations, which allows them to classify all groups of orders less than sixteen. The text concludes with a proof of the insolvability of the quintic, which is a great culmination for students’ introduction to abstract algebra. The treatment of topics in ring and field theory is complete and well-done.

The exercises are well-written and non-repetitive. Each section ends with six to ten exercises and each chapter features review exercises. There are no hints or solutions to the exercises at the end of the text.

Overall, the authors succeed in achieving their goal. However, while it may be advantageous to have a clear text without the additional material that often gets skipped in a first course, there are also some drawbacks. The student is introduced to groups and rings without seeing many of the applications of group theory, and only limited historical information presented. The authors present many of the major results in group theory before presenting the symmetric and dihedral groups, which could be presented earlier to motivate the study of groups and their essential properties. Quotient groups and group actions are often difficult concepts for students to grasp and there are few exercises requiring students to “get their hands dirty” and work inside specific quotient groups or explore group actions on sets.

Thomas Wakefield is an assistant professor in the Department of Mathematics and Statistics at Youngstown State University. His mathematical interests include abstract algebra, representations theory, and actuarial science. His email address is tpwakefield@ysu.edu and homepage is http://people.ysu.edu/~tpwakefield.