All the old standards in a classic remix

This book by Paul Cohn is described as a revision of the second edition of parts of his three-volume Algebra, written in the 1970s for honours mathematics undergraduates. Its coverage of algebra is impressively wide, including most of the "standard" undergraduate topics such as linear algebra, group theory, ring and module theory and basic field theory, as well as less commonly treated topics.

The book also looks beyond the undergraduate realm towards subjects such as functors and categories. Exercises are accompanied by solutions. This encyclopedic book should be a useful reference book for a graduate student or professional mathematician.

Perhaps the closest comparable book is Serge Lang's Algebra, which has also undergone at least one major revision. Its coverage is more sophisticated than Cohn's, with a more advanced development of subjects such as Galois theory and number theory.

I recommend this book for strong undergraduates, graduate students and anyone wanting a good source of ideas on how to present the book's topics, especially its exercises. But it would probably be too challenging for most undergraduates.

Books such as John Fraleigh's First Course in Abstract Algebra provide a gentler introduction. And Cohn's coverage of linear algebra is too lacking in geometric aspects to be completely satisfactory, even though it gives applications such as linear programming.

One quibble: the diagrams are typeset using very different fonts from the text. In the era of TeX, attainable standards in typesetting mathematics are high and commercial publishers have no excuse for such failings.

Andrew Baker is reader in mathematics, University of Glasgow.