I am going to start learning Abstract Algebra soon. I was originally going to start with Dummit and Foote, but I am starting to abandon that idea. I want to use a "hardcore" algebra book. I don't mind how terse a book is (I actually quite enjoy this), I can work my way around that so it isn't an issue. I just want a book that is used by the hardest grad schools, essentially the highest level algebra book there is, also one that covers perhaps the most material. To my understanding, Lang's graduate Algebra seems to fit this description, but I was wondering if you all could either recommend what you believe to be better, or confirm what I have heard. Thanks in advance for any response.
[Math] Hardcore Abstract Algebra Book Request
abstract-algebrabook-recommendationreference-request
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I found Aluffi's book to be a refreshingly gentle advanced text on abstract algebra, Dactyl. It covers algebra in just the right amount of detail for a graduate course-not spoon-feeding, but not ultra-abstract or terse either-with a TON of good exercises. And it really tries to get students comfortable with using categories and commutative diagrams,too.
Unfortunately, my one beef with the book is that it's very classical. It doesn't cover a lot of topics that are very relevant today-such as commutative algebra beyond the bare bones, nothing on group representation theory (!), nothing on quantum groups, etc. But then, with the exception of Louis Rowen's awesome 2 volume course-which I strongly recommend as collateral reading-none of the standard graduate algebra texts do.
However, there IS something unique and cutting edge in the book that I don't think Aluffi gets a lot of credit for-there's a whole section on the elements of homotopic algebra, which is a very significant topic of current research interest and I don't think it's covered in any of the other standard graduate algebra books! But other then that, it covers the run of the mill stuff. It does it very well, but it's not very original in content with that one major exception. However,considering how well Aluffi presents the rest of the material, that one major exception makes the book worth having.
Overall, it's a VERY nice book indeed and supplementing it with Rowen's books or Atiyah/Macdonald and Reid and Paul Etingov's upcoming book on representation theory will give you everything you need for a terrific self-study course in algebra that covers all the essentials.
Several comments: I think that most "easy" abstract algebra books are well worth reading if you find an intermediate or advanced text too dense. I am not aware of any abstract algebra that is not "the real deal".
I wonder what the rest of your mathematics background is. My answer here would be affected if, say, this was the first "real" math course you ever took, or on the other hand if you had attended courses like real analysis or topology before this class.
If this is your first "real" class, there is the possibility you just haven't finished your habilitation phase moving from "calculus" type courses to "proof" type courses. For the first 12 years of school, US students are taught a lot of things in the mathematical curriculum which do not really represent mathematics properly. It is often a shock to adjust to the real thing. I would hope you might consider trying abstract type courses again, in this case :) Sometimes it takes getting used to. If this is the case, I don't know if you have the experience to say that the way things are taught should change.
If you have had "real" math classes before, then it's still possible you just don't have a very pedagogically oriented professor. It is often very hard to change the way one teaches, and several profs don't have the patience to do it. If you think your prof is pretty open, then it would be a fantastic idea to go talk to him about how the course went. He would probably be very happy to see you take such an interest in it. It might not result in an algebra revolution at your school, but it might make it a bit better for your peers!
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Hungerford is great. You should prove the things he puts out as exercises. It will give you a strong grasp of Algebra. I used it to pass the Ph. D. qualifier at the University of Texas.