Rating: Summary: Great reference, poor text Review: If you want to learn Algebra, get Artin or Shafarevitch (encyclopedic and much more understandable) or even van der Waerden. Lang does not write in a manner that's understandable by novices and is, in fact often obscure even for cognescenti. His definitions are often couched in the most abstract of category theoretical forms. Amazingly, after doing such, he spends his chapter on homological algebra sneering at such abstractions (He calls it "Abstract Nonsense"). Also, be aware that the change in "editions" seems to consist mostly of adding extra exercises to fill out the page. Once you know the material, Lang's book becomes an excellent reference - even there, I'd recommend Bourbaki or the Russian "Encyclopedia of Mathematics" for full expositions.
Rating: Summary: This will teach you how to run if you know how to walk Review: Lang's algebra book is one of the best algebra books available today. I agree with what most other readers have said. Namely, this shouldn't be your first foray into the subject, the proofs are often terse and take a good amount of time to absorb and there is a conspicuous lack/obscurity of examples. To cite an example, he gives a non-singular projective group variety as an example of a certain group. I shall not give an example of a terse proof. Let's just say that it suffices to note that whenever he says something is 'obvious', the non-expert reader should be prepared to scribble on 4-5 sheets of paper if she wishes to understand why it's 'obvious'.The core matter (groups, rings, fields, modules) is the same as that you'd find in any other book. As far as topics are concerned, there are just too many fascinating topics in Algebra to cover in one book - even in one like Lang. He covers a fairly wide assortment of topics though. For instance, he covers most of the commutative algebra one would find in Atiyah-Macdonald. He also has a chapter and half on Algebraic Geometry which provides a good preparation for a treatment of schemes like that in Hartshorne Chapter 2,3. His section on Galois theory is detailed and even gets into Galois Cohomology. His chapter on Valuations gets into the theory of Local Fields, but only just. The chapters on multilinear algebra and representation theory are fairly detailed. I talk about the section on Homological Algebra later. Regarding category theory, Lang likes to phrase his definitions in the language of category theory for a reason. It's much much better this way. Category theory is an elegant way of describing some commonly occuring themes in Mathematics, particularly algebra. His preliminary section on category theory provides a good foundation to study the rest of his book. Another advantage of using category theory is that this prepares the reader well for further study in Algebraic Geometry and Algebraic Number Theory where the language of category theory is ubiquitous. On a related note, the book contains all the homological algebra necessary to read Hartshorne's Algebraic Geometry which is indeed quite wonderful for the reader who's not prepared to fight through Eisenbud's encyclopedia on commutative algebra. One of the other reviewers mentioned that Lang sneers at categorical arguments by calling them 'abstract nonsense'. This isn't quite right. He does call them 'abstract nonsense' but not because he dislikes them or harbours any sort of negative feeling towards them. Rather, he does it because the term 'abstract nonsense' is the common and accepted name used to refer to such arguments. Indeed, it's roots can be traced back to Steenrod who was one of the founders of the subject.
Rating: Summary: Poor Text, Poor Reference Review: The only thing that comes through cleary in this book is the author's arrogance. The reader will get a very good sense that Lang understands the material, but the reader will also get the sense that there must be a better book out there. Even as a reference, this book is lacking. The index is incomplete, and many important concepts are either poorly defined or the relevant equivalent definitions are not given. Finally, the order in which the material is presented is haphazard at best. This is truly a text only for those with a Ph.D. who work primarily in the field of Algebra.
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