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Question is as it is stated in title.

I will be joining for PhD program in this July 2014.

I am interested in working in Algebra/Algebraic Geometry/Algebraic Number Theory.

I tried to learn algebra from Serge Lang's book (some two and half years back), but due to lack of background, I could not understand a bit of it, and I lost interest.

I always wanted to read it, but because I could not understand anything in it, and because most of my seniors keep saying "Lang is difficult," I lost interest and hope in reading that. I easily get irritated by seeing that book.

One of my friend gave me his copy of Abstract Algebra by Dummit and Foote. It was totally different from Lang, and I was comfortable reading that. Now I have done almost all exercises in three fourths of the book (with help of MSE).

The curriculum for coursework in the coming year is:

  • Review of field and Galois theory: solvable and radical extensions, Kummer theory, Galois cohomology and Hilbert's Theorem 90, Normal Basis theorem.

  • Infinite Galois extensions: Krull topology, projective limits, profinite groups, Fundamental Theorem of Galois theory for infinite extensions.

  • Review of integral ring extensions: integral Galois extensions, prime ideals in integral ring extensions, decomposition and inertia groups, ramification index and residue class degree, Frobenius map, Dedekind domains, unique factorisation of ideals.

  • Categories and functors: definitions and examples. Functors and natural transformations, equivalence of categories,. Products and coproducts, the hom functor, representable functors, universals and adjoints. Direct and inverse limits. Free objects.

  • Homological algebra: Additive and abelian categories, Complexes and homology, long exact sequences, homotopy, resolutions, derived functors, Ext, Tor, cohomology of groups, extensions of groups.

  • Valuations and completions: definitions, polynomials in complete fields (Hensel's Lemma, Krasner's Lemma), finite dimensional extensions of complete fields, local fields, discrete valuations rings.

  • Transcendental extensions: transcendence bases, separating transcendence bases, Luroth's theorem. Derivations.

  • Artinian and Noetherian modules, Krull-Schmidt theorem, completely reducible modules, projective modules, Wedderburn-Artin Theorem for simple rings.

  • Representations of finite groups: complete reducibility, characters, orthogonal relations, induced modules, Frobenius reciprocity, representations of the symmetric group.

The suggested book for this is S. Lang, Algebra, 3rd Ed., Addison Wesley, 1993.

I do not know if I have to choose some other book or convince myself (I do not know how) to be with Lang's book. I want to remind again that I have no motivation to.

Another thing that I heard is that it is better to use Lang as reference book than a textbook for a course.

I am in a confused state.

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Just take the class, read the book, go to all the lectures, and ask questions about the parts you don't understand. If you really don't want to read Lang, don't. Dummit and Foote is a good book too. –  nomen Jun 19 at 5:56
@nomen I read most of dummit foote and believe it is time to move on to some other book (a bit advanced).. Thank you for your suggestion :) –  Praphulla Koushik Jun 19 at 6:06
Any specific reason for downvote? –  Praphulla Koushik Jun 19 at 6:16
I'm really hoping that you read Lang, understand it all, and then turn around and write a book that is better organized (don't reference later chapters) and more accessible (present requisite knowledge before assuming it) with what you've learned. I'd buy it. –  dotancohen Jun 19 at 8:40
@dotancohen : I am happy for what ever you mean.... :) :) –  Praphulla Koushik Jun 19 at 8:44

3 Answers 3

up vote 10 down vote accepted

I really disliked Lang's book, at first. His depth is comprehensive, but he jumps into most topics at the maximum level of complication and detail, which is extremely jarring for someone just learning the material. On top of that, the topics are sometimes somewhat out of order, with examples from one chapter coming from material from a later chapter (though this can be said of Dummit and Foote as well). It can be very frustrating trying to learn anything from this book if you've never seen it before.

(By the way, what I have heard is that Lang wrote the book by dictating it to his secretary. I'm not sure whether or not this is a rumor, but I wouldn't be surprised if it's true. The one or two sentence paragraphs, the sometimes zig-zaggy flow - it seems like exactly what would happen if he came back to his office after giving a lecture, put his feet up on his desk, and churned out a section orally, stream-of-conciousness style.)

With this said, when trying to study the same topics for my course from other books, I found myself eventually coming back to Lang. Other sources proved to be good introductions to a subject, but few approached the depth that Lang eventually got to. I eventually came to really appreciate the vast array of examples Lang included, even though some of them were above my head at first. You could do what I did, too: read other books when first learning a topic, get a basic grasp of it, then transition to Lang to finish. However, it can be hard to keep terminology straight during these transitions, so the best approach may be to just work through Lang slowly, realizing that you won't be able to understand right away and that that's okay.

Anyway, if you do choose to replace Lang with something else, it shouldn't be Dummit and Foote. Dummit and Foote is a great book, and you'd do well to read it concurrently, but it's a level below Lang. Many of the topics you listed will only be touched upon briefly, and some not at all. A common alternative is Hungeford, which achieves the same level of detail. Personally, I am partial to Isaacs. There are a couple topics (such as Hilbert's Theorem 90) which are not treated quite as thoroughly in Isaacs as they are in Lang, but for the most part, it is on the same level, and I find Isaacs exposition to be very clear and well-motivated.

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+1 for "It can be very frustrating trying to learn anything from this book if you've never seen it before." I do not want to replace lang with dummit foote.. :) –  Praphulla Koushik Jun 19 at 8:47

You have to read whatever you are comfortable reading. There is no reason to torture yourself.

At the same time, there is also no reasonable reason for you to put yourself in a position where you eliminate one of the possible sources of information about what you want to know! (Lang's book is not only known for being difficult: it is also known for being an amazingly good source of information.)

As you go along, you will find yourself in the situation where you want to know something and you have to know it, and yet the only source available is quite not what you would have hoped it to be —this will happen with papers, for example. What are you going to do? Quit the subject because you do not like the source? Wait for somone to provide another source (as this may well never happen, this may have the same consequence as quitting the subject!)? You will have to make do with what you have. Yes: that takes some training... and you might take the opportunity of this course to train yourself also in that respect.

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+1 for "you will find yourself in the situation where you want to know something and you have to know it, and yet the only source available is quite not what you would have hoped it to be"... I strongly believe this would happen in near future and that is the reason i have asked suggestion.. It would be useful for me if you share what was your experience in that case... –  Praphulla Koushik Jun 19 at 6:38
My experience? I just read whatever I had to read. –  Mariano Suárez-Alvarez Jun 19 at 6:55
If you read what you have to read "before" you are ready to read and you get irritated by it... then?? –  Praphulla Koushik Jun 19 at 7:00
You learn to manage your irritation, what else? There is nothing specific to mathematics here, really. –  Mariano Suárez-Alvarez Jun 19 at 7:04
fine fine... there is nothing specific to mathematics... –  Praphulla Koushik Jun 19 at 7:17

First of all, congratulations on entering the PhD program. The particular choice of book really doesn't matter that much; there certainly isn't a single, canonical source for any of the items on the syllabus you posted. Figure out the way you learn best, including what kinds of textbooks you like, and go from there.

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