Abstract Algebra Book Reviews - Best Abstract Algebra Textbooks

One piece of advice for anyone studying, teaching, or working within the field of abstract algebra: do not rely on any one single book. Different books have different styles, strengths, and weaknesses. On this page I've done my best to highlight these.

Two pieces of advice for students: do not assume a book is poorly written just because you don't initially understand it, and revisit books over time after taking a break from them, learning more, or reading other books. Some of my favorite books are ones that I had trouble understanding the first time I picked them up.

One piece of advice for teachers: remember how hard abstract algebra was the first time you learned it. I'm someone who considers himself extraordinarily gifted in the area of abstract algebra. It comes naturally to me, I have a good intuition for it, and I find I can even often visualize things in it. But I still find it very tough.

Abstract Algebra by Dummit and Foote is one of my favorite abstract algebra textbooks, the one I would recommend to use if teaching an undergraduate first course in abstract algebra. However, it covers advanced topics and will likely be useful for grad students as well; I first discovered it in grad school and found it immensely useful.

Strengths: Relatively slow pace, clear explanations, and thorough coverage of all foundational topics makes this book well-suited for self-study. It is an excellent book for an introductory undergraduate level, but it goes so deep that it is quite useful for more advanced study, including at the graduate level.

Possible downsides: Many of the exercises in this book will seem easy or too straightforward to advanced students. A student with extensive exposure to tough, abstract math courses, studying abstract algebra for the first time, may find it too easy, especially in the earlier chapters.

Complemented by: Advanced students, or students who want more challenge or a leaner, more general approach, may wish to check out the Isaacs, which speeds through the abstract development of the subject, or the Lang, which is also more advanced and includes some more interesting examples of many structures.

Algebra by Saunders Mac Lane and Garret Birkhoff, is a modern update of an old classic Algebra text by the same authors, A Survey of Modern Algebra.

Strengths: By starting with basic arithmetic, this book provides a more concrete and gentle transition into abstract structures that will make abstract algebra perhaps seem a bit more natural to some students. The book also more closely follows the historical development of the subject. This book is broad in focus, reflecting MacLane's breadth and holistic approach to mathematics. It explores many topics omitted in most introductory algebra texts, including bilinear forms, lattices, tensors, multilinear algebra, and even a bit of category theory (which MacLane is famous for).

Potential downsides: This book is rather middle-of-the-road in terms of level of explanatory detail and style. Some students may prefer books that more slower, go into more depth, and provide more examples, like the Dummit and Foote, whereas others may prefer books like the Isaacs which are more abstract, or books like the Lang which are more advanced and omit more details.

This book is oriented towards undergraduate math majors. It focuses exclusively on groups and group theory, and emphasizes geometry (particularly, symmetry) and intuitive ways to visualize group structures.

Strengths: Excellent for developing intuition, encouraging experimentation, and visualizing abstract structures. Clear writing. Challenging exercises that require genuine creativity. Goes remarkably deep for how brief and how accessible it is.

Potential downsides: This book only covers groups, and does not touch on rings or fields, so it alone is not sufficient for a thorough introduction to all the basic topics of abstract algebra. Not terribly useful as a reference text.

This book is exceptionally dense, but remarkably easy to read given its advanced nature. It is usually used at the graduate level. This book changed publishers to the AMS series (blue and yellow cover) as it went into a newer edition. I have linked to both editions below. The newer edition is more affordable, and is much larger and has much clearer typesetting.

Strengths: Exercises are diverse in difficulty level, and very helpful for deepening understanding.

Possible downsides: The style of proofs includes tedious explanations of certain steps and details which I find intuitively obvious, and which I would prefer be omitted or left to the reader. The book is also very sparse in concrete examples, and frequently introduces abstract structures and then proceeds to reason about them extensively without ever giving any examples of them.

Complemented by: This book is complemented by the Dummit and Foote at a more introductory level, and by the Lang at a more advanced level. Both books provide rich, concrete examples which fill the void set up by this book.

This book is notoriously tough to work through, but I find it can be immensely valuable for people already familiar with the subject, and looking to deepen their knowledge. I recommend this book to people who have already read at least 25 other abstract mathematics books. Yes, at least 25.

Strengths: Clearly written, provocative, omits tedious proofs, but still engages in much discussion. Unusual and unique in choice of topics and presentation. Rich and diverse concrete examples are given for each abstract structure introduced. Rich connections made to other areas of mathematics. Quirky and creative exercises.

Possible downsides: Advanced text, fast pace, assumes both prior knowledge of abstract algebra and fairly high level of mathematical sophistication. Completely omits any discussion of lattices.

This book is best known as being a preparatory book or supplemental explanatory text for the impenetrably dense Algebraic Geometry by Robin Hartshorne (also shown below). Algebraic geometry is a subject based on commutative algebra, which is an advanced subject requiring extensive background, and appropriately, this book requires prior knowledge of most of the basic structures of abstract algebra, including groups, rings, fields, and vector spaces.

Strengths: Rich prose, emphasizing history and development of the subject, development of intuition, and motivation for mathematical structures which are often seen as painfully abstract.

Potential downsides: Much better-suited for self-study than an a textbook. Does not read linearly, and requires one to jump around frequently, often re-reading material many times, in order to understand it thoroughly. Some people find this book too wordy, although I do not mind this.

Lattice theory is a sub-discipline of abstract algebra which is little-known, and I think greatly under-appreciated. This book is what I think of as the authoritative volume in lattice theory. It goes quite deep, and includes a lot of cutting-edge results and countless references to the primary literature.

Because Lattice theory is such an accessible subject, this is one of the few mathematics books that is accessible enough to serve as an introductory text, yet goes deep enough that it can take people very close to the forefront of academic research.

Strengths: Lucid and captivating writing. Helps to develop a clear intuition for the subject. Huge numbers of exercises and problems at the end of each chapter, and problems are both fun and illuminating, and sometimes very deep and open-ended, characteristic of the sort of thinking needed in research.

I recommend for most serious mathematicians to at least borrow and browse this book and work the first two chapters. I found it helped greatly to develop my intuition for the many situations in which lattices arise, especially in set theory and number theory (as with lattices of divisibility).

Potential downsides: I cannot think of any downsides to this book other than that its subject matter is rather esoteric.

You may be surprised to see a book on analysis (advanced calculus) in a list of abstract algebra books. But this is no regular analysis text. A full quarter of this book is dedicated to algebraic structures.

The pace of this book's algebra section is fast; it covers groups, rings, fields, matrices and more in a single chapter, and then delves into categories, algebraic structure of the real numbers, and has further chapters on linear algabra, convexity, boolean algebras, and logic.

I recommend this book for self-study for all beginning graduate students in mathematics, and for purchase as a reference text for any mathematician. It is a unique book that I found revolutionized how I think about virtually all areas of mathematics.

Strengths: The only introductory book that I know that introduces algebraic structures primarily from the perspective of analysis. This book's presentation of many algebraic structures is unique and completely unlike that I have found in any other book. The connections to mathematical logic and topology, and the way the book discusses the addition of topological structures overtop algebraic structures are also unique.

Potential downsides: This book presents so many examples, so much material, and so many different ways of looking at things that it is easy to get lost and overwhelmed while reading it, even in the more accessible chapters. Although this book is exceptionally accessible for how deeply and thoroughly it develops subjects, it is still dense and often requires great thought and reflection to understand. This is not a thorough or comprehensive algebra text: it can enrich your understanding of algebra, but it is not a substitute for other abstract algebra texts of the sort covered above.

The links on this page are for Amazon. Often, Amazon can be a good place to buy books both new and used--the Amazon marketplace feature highlights third-party sellers offering used copies. I also recommend checking out Half.com if you're going to buy one of these books here.

I recommend being cautious, however, when buying used books. Check the advertised condition, and make sure to hold the seller to the specifications for the condition--if a book is marked "like new" and it arrives with highlighting or heavy wear, complain and demand a refund or compensation. It's important to hold sellers to high standards so that they don't push the boundaries of listing items in better condition than they actually are!

Overall though my experiences buying through Half.com and the Amazon marketplace have been good. Check the seller's rating, and you will not usually have many problems. You can save a ton of money this way.

There's nothing like holding a hard copy of a book in your hands and working through it, but if you are looking for something free and downloadable, I'd point you to these useful resources! These free e-Books can be downloaded onto an e-reader, printed, or viewed on a computer or tablet.

• Free Abstract Algebra Texts by Robert B. Ash - Robert B. Ash, Downloadable Lectures and textbooks from Intro Abstract Algebra through more advanced topics.
• Abstract Algebra - Wikibooks, open books for an open world - An open-source, wiki textbook on abstract algebra (abstract algebraic systems), an advanced set of topics related to algebra, including groups, rings, ideals, fields, and more.