For those seeking a sound introduction to abstract algebra, or a handbook on the basic ideas, this third edition of Nicholson’s book is highly recommended. It is primarily intended for use on upper-level undergraduate and graduate courses, and is described as a ‘self-contained, self-study text’, characterised by the following features:

- The inclusion of historical commentary and biographical vignettes.
- Five hundred worked examples
- Fifteen hundred graded exercises, together with answers, hints and/or explanations for about a quarter of these.
- Applications to real-world problems.

The first half of the book introduces the fundamental aspects of groups, rings, polynomials and integral domains, the presentation of which is certainly user-friendly. The remaining 250 pages are devoted to more advanced ideas such as splitting fields, finite fields, p-groups, Sylow theorems and Galois theory. Also, for this third edition, there are new sections covering free, semi-simple and projective modules, semi-direct products and the Wedderburn-Artin theorem, taking the reader well beyond the introductory stages of abstract algebra.

The accessibility of the treatment emanates mainly from the features listed above. Firstly, each of the eleven chapters begins with a brief historical overview of the appropriate theme, often followed by additional biographical observations and quotations. Secondly, the introduction of new ideas is always well motivated, usually by linking them to the reader’s previously acquired knowledge or to further historical notes. Thirdly, there is the abundance of examples and well-structured exercises, interspersed with well-written explanatory dialogue. And last, but not least, there is the fascinating nature of the mathematical material itself, with which this book is fully packed.

As for ‘applications’, these mainly appear at the end of the chapters on Integers and Permutations (Cryptography), Groups (Binary and Linear Codes), Fields (Cyclic and BCH Codes) and the chapter on p-Groups (Combinatorics). But reference to the geometric importance of group theory is restricted to a few pages on symmetry groups of simple geometric figures, yielding little insight into algebraic hierarchies such as Eucl(n), Aff(n), PGL(n), and so there is no discussion of group actions on geometric structures.

Although the author describes this book as covering standard material, I feel that most readers will find something new in its five hundred pages. In fact, there is enough here on which to base several courses, and the author has provided a flow diagram indicating chapter inter-dependence, thereby suggesting eight truncated pathways through the book. For example, one such choice is:

Ch 2 Groups → Ch3 Rings → Ch4 Polynomials → Ch 11 Finiteness Conditions.

However, the claim that the book is ‘self-contained’ is partly vitiated by its coverage of vector spaces, which is just a compressed revision of the fundamental aspects. Also, many who are new to the later (more difficult) material may wish to do what we, as teachers, always recommend; that is, read a variety of texts, look at different proofs and observe variations other differences.

Finally, although the historical commentary is necessarily compressed, it is occasionally misleading. For example, in the introduction to chapter 4, it is said that ‘symbolic algebra in the form we know it today developed in Arabia between 600 and 1000 AD’ — a statement which could posthumously incite the litigious expertise of François Viète and the pugnacious tendencies of René Descartes.

As one would expect with a third edition, I was hard-pushed to find errors or misprints in Keith Nicholson’s book; although I did spot one or two things that may cause head-scratching amongst autodidacts. For instance, there is some ambiguity of notation on p159, where kR is defined as the *set*, which makes the subsequent equation kR = 0 seem like undue poetic licence. Clarification is also required concerning the comment on p. 162 indicating the total absence of examples of non-commutative division rings — despite the subset of invertible matrices in M_{2}(R) mentioned on the previous page. But, when I recall that, in my student days in the early 1960s, the recommended texts were the likes of Birkhoff and MacLane and Zassenhaus’ book on group theory, such observations seem churlish considering the overall excellence of this student-friendly book.

Peter Ruane has now escaped the bureaucratic confines of higher education, where he spent a working life training future primary and secondary mathematics teachers.