Is it bad to keep aside Lang's Algebra in graduate school?

manpreet Best Answer 3 years ago

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.