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.
Best Answer
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.