Algebra

"Lang's Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books." NOTICES OF THE AMS

"The author has an impressive knack for presenting the important and interesting ideas of algebra in just the right way, and he never gets bogged down in the dry formalism which pervades some parts of algebra." MATHEMATICAL REVIEWS

This book is intended as a basic text for a one-year course in algebra at the graduate level, or as a useful reference for mathematicians and professionals who use higher-level algebra. It successfully addresses the basic concepts of algebra. For the revised third edition, the author has added exercises and made numerous corrections to the text.

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.


This is the standard reference for algebra   September 6, 2000
 8 out of 13 found this review helpful

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.


The way to learn algebra   July 18, 2006
 6 out of 10 found this review helpful

Most critisism of the book are based in the fact that topic are not treated in deep, and the reason is that there is no need to do that. Lang's porpouse is to introduce the reader to HIGHER ALGEBRA, while for example Dummit & Foote just end in Category Theory or Homological Algebra. He just introduce what he consider necessary to know about groups. Of course, if your porpouse is to learn, let's say Group Cohomology, then its good for you to know as much as possible about groups, modules and stuff, but Lang's try to focus on what HE consider is important to be known.
One has to have in mind that its easy to present the basic form of Cohomology or modules as Dummit does, but its has no continuity in the sense that will follow you to nothing, just to know some basic concepts. That's why i disagree with the people that say that is like an encyclopedia. Lang's development in Algebra is AMAZING. Ok, one can argue that it can be stated more ''friendly'' such as Galois Theory by Artin book does. But for Galois theory that's easy while in General Algebra is doesn't. Just take a look at Galois Theory section [which is, as every book, based on Artin works], there is nothing that is understandable in it, and its not an extensive work, because Lang will not USE in his HIGHER ALGEBRA.
The whole thing can be explained if you notice the Algebraic Number Theory book of Lang... you can consider Algebra as a preparation for his real book.
My education in math [academically] is minimal, but i always recommend this book, because when i get a subject i go directly to Lang and everything seems clear. Also, notation is really CLEAR and, comparing with authors like Jacobson, he doesn't mess with [unless necessary] formalizations.
So, my advice:
If you want to LEARN algebra, and work in algebra, READ LANG, READ LANG, READ LANG.
If you want to get some knowledge about algebra, and take a course, this is not your book.