“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.

5 Responses to “Algebra”

1. Mark says:

   October 30, 2009 at 12:00 am

   This book covers a very wide variety of topics in algebra, and has lots of exercises. It is a great book for reference.
2. Cloudy Skies says:

   July 4, 2009 at 12:00 am

   Lang’s Algebra is a very broad reference in abstract algebra: most subfields are at least mentioned, and there is a wealth of detail on many important topics. Overall, Lang’s writing is clear and his exposition makes sense. The sheer quantity of information contained in this book, however, forces Lang into a less-than-logical general organization. His index makes up for that to a large extent, and this book is probably best used as a reference. Of course, Algebra makes an excellent reference and complement to any survey course in abstract algebra, because of its clarity and comprehensiveness.

   That said, there are a few caveats that I would like to mention. Lang’s terminology is nonstandard (eg “entire ring” for what other authors call an “integral domain”), and these accumulated idiosyncrasies become increasingly irritating as one reads through the book. Second, don’t expect many applications. I found this to be a problem in Galois theory in particular, where Lang’s exposition is completely clear and easy to understand, but seemingly pointless. He neglects even the classical and fairly abstract applications, like the Galois theorem, and he leaves infinite-dimensional Galois theory as an exercise. Now, there are not many applications in general in this book, so be aware that the clarity of Lang’s exposition comes at a certain cost (ie, Lang never gets bogged down in too many examples or minor points). Third, Lang occasionally uses notation that he never defines and that does not appear in the index.

   In summary, this is a wonderful general reference for algebra, but not a substitute for more specialized or more elementary textbooks in specific subfields, especially Galois theory. The best way to take advantage of Lang’s clear, abstract exposition is to make a real effort to come up with examples on your own and use Lang for the general case. I wouldn’t rely only on Lang’s exercises either: they don’t cover as much as you might want them to, but the ones that Lang does include are challenging and important.
3. Phi L. Le says:

   May 25, 2009 at 12:00 am

   There was no tracking for my book. I didn’t know what happened. It took over 2 and half weeks to get it. I sent email to ask them about my book situation and got no answer.
4. GaloisGuy says:

   March 11, 2009 at 12:00 am

   This is among the finest references for Algebra I have ever seen. Comprehensive is an understatement. The only point of contention may be the terse language used throughout the text which makes it hard for novices. Great text to have on your shelf!
5. J. Bogaarts says:

   January 8, 2009 at 12:00 am

   This is a great book. The only thing I have to add to the other five star reviews is that there is a web page containing lots of information about the book “wherein can be found corrections, commentary, and divers supplementary material … “. It is authored by George Mark Bergman. Thanks George Mark!!

   Google for “Companion to Lang’s Algebra”.