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Abstract Algebra: An Integrated Approach

Joseph H. Silverman
Publication Date: 
Number of Pages: 
Pure and Applied Undergraduate Texts
[Reviewed by
Michele Intermont
, on
A quick review of these archives alone will show that textbooks for undergraduate abstract algebra courses are not in short supply. Several of them are excellent, and as an instructor, I have an embarrassment of riches in choosing for my course.  I expect this text will be on that list when next I get to teach the subject.
As the title suggests, this text interweaves group theory with rings and fields. The chapters alternate focus: first groups, then rings, then fields, before shifting back to groups again, etc.  (There are some chapters on vector spaces too.) As expected, the beginning chapters include lots of examples as well as definitions.  Maybe not expected is that cosets and ideals are introduced early on. These first chapters also include some meaty results.  Lagrange’s theorem on the order of subgroups is found in the first chapter on groups; the interplay between maximal and prime ideals and rings and fields is found in the first chapter on rings; the existence and structure of finite fields is found in the first chapter on fields.
The author explicitly writes in his introduction that he was not aiming to produce a reference book, but rather a book to be read as one enters the subject. I do think my students will find this very accessible.  Indeed, Silverman won the American Mathematical Society’s Leroy Steele prize for mathematical exposition in 1998 for his graduate texts on elliptic curves; here he confirms his expositional skills.
The book contains more than enough material for a full year course.  After groups, rings, and fields, one finds Galois theory and module theory and then, as the author calls it, the “topic of the week”. These topics include some category theory, come representation theory, and some elliptic curves among others.  A sample syllabus for each of three terms is included at the end of the text (based on 36 class meetings). It would thus be easy to structure a second semester in various ways.  I have strayed away from using a second course in algebra to study Galois theory, but some of my colleagues have not, and this single book would easily allow us to adopt the text without regard to who teaches the second term. That flexibility accounts for some of the heft of the book, as compared to a text such as Gallian’s.  So I was surprised to see how affordable it is – about $50. There are also plenty of exercises at the end of each chapter, easily identifiable with the subsection of the chapter to which they pertain.
Silverman’s dedication says “This one is for the next generation”.  Indeed, this is a wonderful resource for training the next generation of mathematicians.


Michele Intermont is an Associate Professor of Mathematics at Kalamazoo College.  She is an algebraic topologist.