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Good abstract algebra books for self study

By Daniel Johnston
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Last semester I picked up an algebra course at my university, which unfortunately was scheduled during my exams of my major (I'm a computer science major). So I had to self study the material, however, the self written syllabus was not self study friendly (good syllabus overall though).

The course was split up into 3 parts, group theory, ring theory and field theory. As a computer science major we only had to study the first 2.

Now that I passed the exam for this course I want to study the field theory part ( which covers Galois theory, etc).

So, now I want to ask whether any of you know any good books on abstract algebra, which lift off at basic ring theory and continue to more advanced ring theory and to finite fields, Galois theory, ...

Please keep in mind that I am not a math major, and that I would like books which are suited for self study (thus a lot of examples and intuition).

Thanks in advance!

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12 Answers

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There's always the classic Abstract Algebra by Dummit and Foote. Section II of the text gives a nice treatment of ring theory, certainly providing plenty of review for what you have already covered while introducing more advanced concepts of ring theory. Section III will cover the field and Galois theory you're interested in. Some of the exercises can be difficult at times, especially for self-study, but the authors tend to give a number of examples and always provide the motivation for why they are doing what they are doing.

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Fraleigh's "A First Course in Abstract Algebra, 7th Edition" is a good book for self study. It is easy and good for the beginners, and it has a complete solution manual written by the author.

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Try Contemporary Abstract Algebra. This one, I think, has lots of nice examples. The following is from Googlebooks:

"Contemporary Abstract Algebra 7/e provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists, physicists, and chemists. The text includes numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings giving the subject a current feel which makes the content interesting and relevant for students."

Also, I would like to suggest you read this article in wikipedia. You may find the references valuable.

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Note: This answer is copied over from an answer I gave on a previous very-similar question, because it still applies here.

This is likely not going to be a popular suggestion, since it's relatively unknown, but I think the perfect book for you is Allan Clark's Elements of Abstract Algebra.

It's a unique book that covers the basics of group theory, ring theory, and even a tiny bit of Galois Theory, but it does it almost entirely through problems. Every chapter begins with a short section defining some terms and giving a few basic proofs, and then it leads the reader through the rest of the exposition in a series of problems, some difficult, some not. The end result is that if you actually do all the problems, you've written the book yourself. It's impossible not to be comfortable with basic abstract algebra if you take this book seriously.

It's also probably the cheapest book on this entire list :)

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I learned abstract algebra from Rotman's "First Course in Abstract Algebra". His expository style is easy to follow and the exercises he gives are helpful.

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It depends on which subject you’re interested in particular. Here is a list of the books if find go in the greatest depth and yield the clearest intuition on each major subject:

Familiar Structures:

Abstract Structures:

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I am speaking from the standpoint of a student, and I think that a very good book on introductory abstract algebra that doesn't get mentioned very often is Basic Algebra by Anthony W. Knapp. From experience, the text is accessible with very little pre-requisite knowledge, is less "talkative" than Dummit and Foote (and in my opinion, definitely not dry, unlike D&F), and more rigorous in exposition than Artin's Algebra, although Artin's book is a good and standard first text as well.

Knapp covers most basic topics that the undergraduate student needs to know and is largely self-contained. I think, for the first seven chapters of this book, you can't really do much better by way of alternative texts. However, you could supplement or even replace the eighth chapter with Introduction to Commutative Algebra by Michael Atiyah and Ian MacDonald. However, if you are reading algebra for the first time, I don't suggest using Atiyah's book, unless you are feeling very confident or very lucky! :) Having said that, it is an excellent book and you should try reading it at some point. For the ninth chapter, you could use Emil Artin's classic little book on Galois Theory, based on his lectures on the subject. Another good reference which I haven't used but heard quite a few good things about is Nathanson's Basic Algebra: I (Chapter 4 (?), I think). Yet another book on Galois Theory is D.J.H. Garling's Galois Theory, which is where I initially learnt my Galois Theory from. As for Chapter 10 in Knapp, I have nothing to say, since I never got down to reading it.

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One book that I did not see mentioned, but which really deserves some accolades is the recent book Visual Group Theory by Nathan Carter. There are some excellent accompanying videos by Prof. Macauley on his youtube channel. These go really well together.

The biggest trouble I ran into with group theory and abstract algebra was the dizzying set of definitions that most books present at the beginning. You get a bunch of definitions with little or no motivation and with little description of the underlying geometry of how the binary operations work. The nice thing about the Carter book and the videos is that it spends a lot of time working though group diagrams and showing the "symmetry" of a group. It is easy to get caught up in the formalism, but without a good intuitive understanding of how different groups work--and how simple groups differ--it is easy to get frustrated--especially in self study. Also note that the Carter book has exercise solutions at the end.

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My suggestions are 1) Fraleigh 2) Gallian 3) Herstein and 4) Rotman,

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Fundamentals of Abstract Algebra by Malik, Sen & Mordeson is a very good book for self study.The topics are covered in detail with many interesting examples and exercises.Also it provides hints and answers to difficult questions making it suitable for self study.

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There is no easy or right answer. I know brilliant professors who cannot easily decide what textbook to use for an advanced math course, and for good reason. Every book has its own strengths and weaknesses. I suggest you go to your math library (assuming one is available during this pandemic) and examine several books. A book you like might be hated by someone else, it is highly individual. You likely will need at least two or three books so you can go back and forth. Even a good book can be bad in a particular section and vice versa. Use a common textbook that has gone through at last two or three editions as a guide as to what topics to cover and then be prepared to use alternate books to actually learn the topic.

Use the Internet. Don't be afraid to read lecture notes or check Wikipedia. Also, Professor Keith Conrad (Univ. of Conn.) has dozens of expository papers on algebra on his web site, some are easy, some are difficult, and some are advanced or specialized. I have found that lectures by professors at lessor known universities to often be better than those by professors at famous brand name universities. That being said, I have found lectures by Unv. of Berkeley professors to be quite good, and lectures by MIT professors to also be good, but the latter are often very fast paced and better for review than to self learn from as they are so intense.

I suggest you get an easy book, an intermediate book and eventually a hard book. Herstein's: Topics in Algebra is harder than Birkhoff and MacLane's book, but Birkhoff and MacLane's book is good for learning the fundamentals. As an undergraduate I used Herstein, but I think it is too difficult to self study from.

It is critical to learn the definitions and other fundamentals cold and then go on to a more advanced treatment. (One really smart professor basically told me: memorize definitions, but do not memorize proofs, just understand them.) Herstein loves to give problems and results that are hard using elementary methods, but easy using more advanced methods. In my opinion this a bad way to learn, as not everybody is clever at solving hard problems or following highly technical arguments, and I think it is more useful to put one's energy into learning the concepts and theory that makes it possible to eventually easily understand what is really going on, rather than rely on clever technical tricks or manipulations to get a result with no real deep understanding as to what is really going on. Neither Herstein nor Birkhoff and MacLane cover everything a graduate course would cover. Herstein, in my opinion, makes the subject seem more difficult than it is.

A free book is by Robert B. Ash (University of Illinois at Urbana-Champaign), titled: Abstract Algebra: The Basic Graduate Year, it is available as a series of PDF's on his web site.

I also use: Algebra: A Graduate Course by I. Martin Isaacs, it has its strengths and weaknesses. It is is elegant and the proofs are carefully done, but it may be too abstract and condensed to self study from.

A classic is the two volume (mostly of the time only the first volume is used) set by B. L. van der Waerden titled: Modern Algebra. There is at least two English editions (the original is in German). Even though the editions differ, any English edition is fine. And if you really groove on abstraction there is Serge Lang's book, simply titled: Algebra.

Alternatively, audit the class you need. Self study is a mixed blessing, as while there are no exams, required homework or time pressure, it can be frustrating and inefficient.

Good luck. I think it is great you are so motivated. Keep the faith. Don't worry if at times you get overwhelmed or discouraged --- self study is not easy --- it has happened to many of us at some point in time, yet somehow we didn't let it stop us, nor should you.

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My favorite is Introduction to Abstract to Algebra by Keith Nicholson. There is a course in YouTube, by James Cook base on that book.

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