This book is a sequel to the author’s Basic Algebra I, but it is not a continuation: it treats a completely new set of topics, at a much higher level of abstraction. It is “basic” only in the sense that the material studied here is the basis for getting started with modern abstract algebra. The present volume is an unaltered reprint of the 1989 second edition.

The first chapter covers categories, and most of the rest of the book takes a category-theoretic approach. The pace here is more leisurely than it was in the first volume, and the book does not go into great depth on any topic. For example, the chapter on valuations does not go much beyond the ramification index, Krasner’s Lemma, and Hensel’s Lemma. The book does have a large number of exercises, of varying levels of difficulty, and most of these explore further developments and are not simple applications of the text material.

The great weakness of the book is that it is relentlessly abstract: there are almost no concrete examples of the abstract structures that are being studied. Each chapter is careful to credit the areas of mathematics from which it is abstracted (usually these are number theory and algebraic geometry), but we never see any applications back to the source subjects.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.