What better book to kick of my in-depth reviews of textbooks with than than the first math book I bought for myself? In December of 2018 I purchased “A Book of Abstract Algebra” by Charles C. Pinter (and several others), somewhat on a whim, and worked through it. It would not be a stretch to say that this book was the largest factor in convincing me to get a degree in mathematics. What captured my interest about this book was the author’s writing style and how he immediately presents concrete applications of the concepts after he presents them. It felt like peaking in on a whole new world of math which had previously been hidden.
This book is a textbook aimed at undergraduate students taking a first course in abstract algebra, which is a course that almost all math majors (and some science majors) will take. Abstract algebra, not to be confused with elementary algebra which is typically taught in high school, can be best summarized as the study of symmetry and the algebraic structures (groups, rings, fields, etc.) used to describe it. A well prepared student will have taken linear algebra and a proofs course, but this is not strictly necessary. I had only taken up to calculus 2 (integral calculus) and was able to dive in without much difficulty. If anything, going a bit out of order and reading this book before taking linear algebra made me appreciate that course much more than I might have otherwise. The book begins gently with a lengthy introduction and easy coverage of operations. There are appendices which cover most of the needed background material in set theory and the integers. Pinter’s writing has a nice sense of narrative to it which made this book a page-turner. That’s not something you can say about most textbooks.
This book has a strong reputation and is commonly recommended through word of mouth. It was through such a recommendation on a math forum that I heard about it and decided to buy it. I have seen it used as the primary textbook in several courses but more often it seems to be used as a supplement. The book covers all of the standard topics related to group and ring theory, including fields and polynomial rings. The whole book is building up to the last several chapters which present the basics of Galois theory. Galois theory is historically significant, beautiful, and useful in many areas of math, so having a nice path leading up to it was rewarding.
One of the standout features to me was the applied nature of the exercises. Each chapter is followed by several sets of exercises. Some sets include additional exposition which introduce an area of application for the topics discussed in the chapter. Some examples include error correcting codes, finite state machines, and kinship structures. Not only are doing exercises the key to learning the material, they also reinforce why the material is important in the first place. Sometimes the way something is presented in the exposition might seem odd or cumbersome, but then it is encountered in an exercise and it suddenly makes perfect sense why that is the best way to think about it.
Abstract algebra is primarily studied by math majors, but the topic also has uses in computer science, physics, and chemistry. In computer science the subject is used in cryptography and in preventing errors in the transmission of information. Abstract algebra appears most notably in physics with the use of groups in particle physics but it is also in the background of quantum mechanics and the study of partial differential equations. The most common use of groups in chemistry is in the symmetries of atoms and crystals.
The book does have several errors and typos in it, but many of the errors have been caught and are listed online. The paperback print quality is excellent and all of the figures are clearly legible. The book is published by Dover so it is relatively cheap for a math textbook. I highly recommend this book if you are interested in mathematics or theoretical computer science. I would be cautious about recommending this as a gift for someone unless you really know that they would enjoy this sort of book. It is an excellent textbook, but might not make for the best light reading.