Also see our review of Volume 1. These two volumes are translations of two Italian-language volumes, *Esercizi scelti** di Algebra*, published in this same series in 2017 and 2018. The exercises are exam questions from the undergraduate program at the University of Pisa.

The background and intentions of the book are described in detail in the Preface. In particular they note (p. ix): “attending lectures and memorizing the statements and proofs of key results are not enough. What one needs to do is apply the theory one studies to concrete examples”. They emphasize that these exercises are not drill but require both some mastery of the theory and some ingenuity.

The two volumes are split up the same way the courses at Pisa are split. The first course, titled “Arithmetic”, and the first volume deal with background material such as proof by induction, some combinatorics

, and the beginnings of abstract algebra. The second course, titled “Algebra 1”, and the second volume deal with the details of groups, rings, and fields. The subject matter in this second volume is very traditional; they cite Herstein’s

*Topics in Algebra* and Michael Artin’s

*Algebra* for further study.

The first fifty pages (about one-fifth of the book) are a concise summary of group theory, ring theory, and field theory (including Galois theory). These are intended to establish the necessary prerequisites for the exercises. Group theory covers homomorphisms, quotient groups, the homomorphism theorems, and Sylow groups. It also has some less traditional material on actions and counting (the orbit-stabilizer theorem and Burnside’s Lemma). The ring theory section has a number-theoretic flavor and is primarily about ideals, but also with some material on ring specializations such as integral domains, unique factorization domains, and euclidean domains. The field theory section is mostly about Galois theory. It uses the language of linear algebra to describe field extensions, and seems to assume this is already known to the reader. Some of the exercises use finite fields, but I was not able to find a description of these in the theory section.

This is followed by a set of “Preliminary Exercises” (about forty pages), that are recommended to be worked before tackling the main exercise section. Each exercise is followed immediately by its solution. There’s not a sharp distinction between the Preliminary Exercises and the main exercises, except that the Preliminary Exercises seem to be easier.

The middle of the book is the statements of the exercises (thirty-five pages). In general, these take a very concrete approach to abstract algebra, asking for properties of specific groups, rings, or fields, rather than proposing general theorems. These are not drill and are not easy, but they do not develop the theory any further.

The last part of the book (160 pages) gives full solutions to the exercises in the middle part. These are written out in detail and do not require the reader to fill in much.

The book includes an index, although it only indexes the Theory part and not the exercises.

My gold standard for problem books is Pólya and Szegő’s *Problems and **Theorems in Analysis I* and *Problems and Theorems in Analysis II*. It may be unfair to compare the present book to these, because their objectives are very different. Pólya

and Szegő

have you develop and learn the theory through sequences of problems, while the present book assumes you have already learned the theory and need some practice in applying it. Still, I feel uneasy about the approach in the present book. I think that working mathematicians do not spend much time on these kinds of exercises, and if they do need such a result they can often look it up or ask a computer. Therefore I would not spend much time in a course on this kind of activity, or give students big rewards for being able to do it. I would not recommend this book as a text or for self-study, because it is too specialized. That being said, I think it can be useful to the instructor as a source of exam problems.

Allen Stenger is a math hobbyist and retired software developer. He was Number Theory Editor of the Missouri Journal of Mathematical Sciences from 2010 through 2021. His personal web site is allenstenger.com. His mathematical interests are number theory and classical analysis.