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I was wondering if some one here would please recommend an abstract algebra text book. In particular, I seeking a text from among one of these:

1)Abstract Algebra by Dummit and Foote

2)Algebra by Michael Artin

3)Topics in Algebra by Herstein.

I was wondering which text is best for a high school student and has great exposition and good problems. My goal is study some algebraic number theory some time later in the near future as I liked elementary number theory.

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Dummit and Foote is a great text, with many examples and diagrams, but the sheer amount of information can be a little overwhelming to someone just getting started. I would probably check out Herstein first for a friendlier introduction. –  Tarnation Sep 18 '12 at 4:54
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Of those three I’d be inclined to go with Herstein. It’s a little old-fashioned, but that’s not really a problem for a first text. The exposition isn’t flashy, but it’s very solid, and there are lots of problems, including some very challenging ones. (Incidentally, when I used it some years ago in a university undergraduate algebra course, my best student by a significant margin was a high school student.) –  Brian M. Scott Sep 18 '12 at 5:15
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My favorite Algebra book is Algebra by Thomas Hungerford. It's a pretty big encyclopedia of algebraic concepts. –  emka Sep 18 '12 at 5:33
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Patience, Squid, it has only been an hour. Fermat had to wait 350 years for an answer! –  Gerry Myerson Sep 18 '12 at 5:54
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@EMKA: Tom’s graduate text? It is indeed an encyclopedia, and quite useful as a reference, but I’ve a pretty low opinion of it as a textbook, especially for a beginner. –  Brian M. Scott Sep 18 '12 at 7:46

4 Answers 4

up vote 5 down vote accepted

To repeat what I said in the comments, my first choice among the three books that you mention is Herstein’s. It’s a little old-fashioned, but that’s not really a problem for a first text. The exposition isn’t flashy, but it’s very solid, and there are lots of good problems, including some very challenging ones.

Having said that, though, I strongly recommend that you spend some time looking at the early parts of all three before you decide, in order to see how well each of the expository styles works for you. This includes the way the author puts words together, the amount of detail in proofs, the number and placement of concrete examples $-$ even the typographical layout, if that affects the book’s readability for you.

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+1 for typographical layout. This seems to be greatly overlooked, but I think the visual imprint and accessibility make a big difference in absorption and retention. –  Andrew Feb 12 '13 at 17:00

My favorite Algebra book is Algebra by Thomas Hungerford. It's a pretty big encyclopedia of algebraic concepts. – EMKA

I would suggest this text if it's your first abstract class. If this is graduate level, I would suggest Advanced Modern Algebra - Joseph J. Rotman.

I've read both, Hungerford starts off with rings then works to fields with most of groups at the end. Honestly, I think he does a better job at conveying rings than Rotman or D&F.

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Rotman's book is not as good as Hungerford, though the later delved hundreds of pages dealing with ring without an identity. You may compare the exercises and see the difference. –  Bombyx mori Sep 18 '12 at 6:08
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Tom’s Algebra is a graduate text that starts with semigroups and groups and is totally unsuitable. You’re thinking of his undergraduate text, Abstract Algebra: An Introduction, which does indeed start with rings. Unfortunately. But he had in mind an audience that included a lot of prospective high school math teachers, and especially for them he saw a pædagogical advantage in starting with rings. I saw his point, but I still don’t like teaching it that way. –  Brian M. Scott Sep 18 '12 at 7:51

I would suggest less on the choice of the textbook and more on solving problems. You do not learn algebra by reading 700 pages of definitions and other's proofs. Pick up any book and work through 70% of the problems you consider not simply a routine manipulation of algebra (for example, verifying the real numbers is a field). Then you will know:

1) Some seemingly difficult problems can be solved by proving simpler lemmas first.

2) Some problems are not clear how to find the best solution, and may need help from others or check on other reference books.

3) Some problems has deep association with other fields, and may provide motivation for your future study.

In the end, no matter which book you use, the goal is if you encounter a mathematical phenomenon you will immediately know what kind of structure may associated with it. For example, if someone is talking about finite groups acting on a vector space or a set, you will be thinking how the representation can be decomposed like or how the group action's stabilizer is. If you can successfully build up a "personal dictionary" that translates mathematical phenomenon on the one hand and abstract mathematical structure on the other hand, then you achieved your goal in learning algebra. In the end you are going to work on problems not in the textbook, and building up a mathematical structure yourself can be very fulfilling if you find its association with other fields of mathematics.

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Is that an implicit nod to Herstein? –  Squid Sep 18 '12 at 6:16
    
no. I have never read Herstein and did not know Michael or Dummit and Foote very well, though I worked through some exercises in both books. –  Bombyx mori Sep 18 '12 at 6:18

Of the three texts that you have mentioned, Dummit and Foote is the only one I have looked at and it is indeed an extremely good entry level book.

I would also recommend A First Course in Abstract Algebra by John B. Fraleigh. This was the first abstract algebra book I encountered and I found it to be an invaluable reference with many useful examples.

As others have said above, the important thing about learning abstract algebra (or any kind of maths for that matter) is doing exercises, rather than just simply reading page after page of theory. This is the way that you learn what is $\textit{actually}$ happening!

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