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Title: Algebra by Serge Lang ISBN: 0-387-95385-X Publisher: Springer Verlag Pub. Date: 08 January, 2002 Format: Hardcover Volumes: 1 List Price(USD): $74.95 |
Average Customer Rating: 4.09 (11 reviews)
Rating: 4
Summary: Excellent if you have the requisite mathematical maturity
Comment: I sometimes joke that "mathematical maturity" is the ability to understand poor exposition. Lang's proofs are often too terse, and even experienced readers will sometimes have to work hard to fill in all the gaps. For this reason this book is not the best choice for most beginning graduate students. Nevertheless, time and time again in my study of algebraic number theory and algebraic geometry, when there has been some nugget of algebra that I had forgotten or never learned, I have found it in Lang and not in other standard texts. So for me, this book is an indispensable reference. Lang also has a knack for giving insightful summaries of advanced topics. Most other authors will at most mention an advanced topic without really telling you anything about it, but Lang actually gives useful introductions to a large number of topics of current research interest.
Rating: 5
Summary: This is the standard reference for algebra
Comment: Concise but comprehensive, Lang's book really has no peer as a reference text for algebra. Both Cohn's and Jacobson's books omit far too many topics. Jacobson's book, which I am more familiar with, follows a very linear structure, which I find limits its appeal both as a reference text and as bedtime reading. This book ought to be challenging for any undergraduate, and perhaps even for some graduate students, but much of the material is essential. The book, unfortunately, contains few examples. This is especially problematic in the section on homology theory, where the abstraction becomes nearly overwhealming. Working concrete examples such as Ext, Tor, or the (co)homology of groups into the text would have been helpful.
Rating: 5
Summary: This will teach you how to run if you know how to walk
Comment: 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.
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Title: Counterexamples in Analysis by Bernard R. Gelbaum, John M. H. Olmsted ISBN: 0486428753 Publisher: Dover Pubns Pub. Date: 04 June, 2003 List Price(USD): $14.95 |
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Title: Algebraic Topology by Allen Hatcher ISBN: 0521795400 Publisher: Cambridge University Press Pub. Date: 15 November, 2001 List Price(USD): $32.00 |
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Title: Complex Analysis (4th Ed) (Graduate Texts in Mathematics, 103) by Serge Lang ISBN: 0387985921 Publisher: Springer Verlag Pub. Date: 15 January, 1999 List Price(USD): $69.95 |
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Title: Counterexamples in Topology by Lynn Arthur Steen, J. Arthur Seebach ISBN: 048668735X Publisher: Dover Pubns Pub. Date: 22 September, 1995 List Price(USD): $11.95 |
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Title: Real and Complex Analysis by Walter Rudin ISBN: 0070542341 Publisher: McGraw-Hill Science/Engineering/Math Pub. Date: 01 May, 1986 List Price(USD): $150.55 |
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