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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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    $\begingroup$ I have converted the question to community wiki, as it's asking for a list of suggestions and there is no single right answer. $\endgroup$ – Zev Chonoles Aug 1 '11 at 0:39
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    $\begingroup$ I think Topics in Algebra by I.N. Herstein is an excellent algebra text. $\endgroup$ – Amitesh Datta Aug 1 '11 at 1:39
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    $\begingroup$ @Amitesh I first learned algebra from the outstanding Herstein and it's awesome exercises, so it'll always have a special place in my heart despite it's old-fashioned approach. $\endgroup$ – Mathemagician1234 Nov 25 '11 at 7:49
  • $\begingroup$ Almost the question was being addressed in sbseminar.wordpress.com/2012/10/25/… $\endgroup$ – Fizz Apr 11 '15 at 19:41
  • $\begingroup$ Also on the topic of Galois theory there's a intro video course by Matthew Salomone. It looks pretty well put together. Actually it reviews enough group theory to even be a decent intro to that topic too. $\endgroup$ – Fizz Apr 11 '15 at 20:31

10 Answers 10

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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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    $\begingroup$ As it contains no answers to the exercises, do you think it is still suitable for self study? $\endgroup$ – sxd Aug 1 '11 at 0:53
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    $\begingroup$ From my experience, most of the exercises are not so difficult that you would need solutions. Those examples that further the development of the theory often either have very good hints or are broken down into smaller, more managable problems (often with hints too!). However, there are solutions (or at least sketches) available on the internet for most of the exercises anyway. It may not be the easiest text available, but I think it is one of the best for a first course. $\endgroup$ – Michael Banaszek Aug 1 '11 at 0:59
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    $\begingroup$ Thanks for your response in the first place. Are there plenty of examples in the book present? This is something where my syllabus clearly lacks! $\endgroup$ – sxd Aug 1 '11 at 1:10
  • $\begingroup$ When I was reading through it, I very rarely found myself wanting an example of a topic or technique and not being able to find one. In general, any time they mention anything they give at least one example of it, though more often than not they'll give three or four. They also constantly provide motivation as to why you are learning any given topic, so very rarely will you ever finish a section wondering why you had to study it. $\endgroup$ – Michael Banaszek Aug 1 '11 at 2:42
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    $\begingroup$ I don't know why,but this book has always annoyed me. It seems like a bloated version of Herstein, except it lacks Herstein's depth and clarity. And the categorical/functorial material seems forced and tacked on. $\endgroup$ – Mathemagician1234 Nov 25 '11 at 7:52
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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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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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    $\begingroup$ +1 for Rotman. ANYTHING by Rotman claiming to be a textbook is outstanding in my experience! $\endgroup$ – Mathemagician1234 Nov 25 '11 at 7:53
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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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  • $\begingroup$ I've never touched this book but based on a review on Amazon by David B. Massey it seems indeed a good intro book to Galois theory. As long as you don't get the horribly executed Kindle version... $\endgroup$ – Fizz Apr 11 '15 at 21:54
  • $\begingroup$ I love that book! $\endgroup$ – abnry Feb 26 '16 at 16:06
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It depends on which subject your 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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My suggestions are 1) Fraleigh 2) Gallian 3) Herstein and 4) Rotman,

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    $\begingroup$ Maybe you could add more information as the poster is asking for a reference request. $\endgroup$ – Ryan Sullivant Jul 2 '13 at 7:34
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    $\begingroup$ I've corrected Rotoman to Rotman - I suppose it was a typo. (I do not know about a book on algebra by Rotoman, neither does Google.) $\endgroup$ – Martin Sleziak Jul 2 '13 at 8:40
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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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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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