Algebra

Welp, that's it. No more comments. I just recommend it after you've read Artin, Herstein, Dummit & Foote. But hey... That is my take

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

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".

When I examined this book as an undergraduate I did not like it; often this is a sign that a book is poorly written, but in this case I just needed more background. Now I see this text as a gold-mine: clearly written, provocative, and rich in examples. I find it refreshing that Lang does not get caught up in tedious proofs (one of my criticisms of Isaacs, another of my favourite algebra texts); anything that is tedious but not difficult, Lang leaves to the reader. Yet the book is not overly concise--a lot of ideas are explained in depth. This book serves as an excellent reference for several reasons. First of all, it's unlike any other algebra book. The choice of topics is unusual; it will certainly expose you to some things you haven't seen before, but at the same time, it is not a comprehensive slice of modern algebra (it doesn't even mention lattices). However, the best aspect of it are the presence of examples, something sorely lacking from most other abstract algebra texts. Whenever a new concept is introduced, Lang presents a variety of examples from material elsewhere in the book as well as other fields of mathematics. These examples alone make this book precious. Although the biggest exercise is just reading and understanding the book, the exercises at the end of each chapter open up a whole other world; they are quirky and creative like the rest of the text. I recommend this book for any serious mathematician to add to their collection. However, it would be waste of time to read it until you already know a great deal of mathematics. This is one of those books that becomes a must-read once have already read 25 or so other serious math books.

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.

I must concur with my fellow readers that in fact Langs Algebra text is extremely dry, the examples are sparse (as compared with, say, Hungerfords Graduate text), readers are left to fill in the gaps which exist within the majority of proofs and, finally, about the exercises; for the most part the exercises abound, they are challenging, non-trivial and in general are extensions of the material, which for whatever reason, have been relegated to the status of mere exercise. But for those who have a 'Solid' foundation in Algebra, preferably at the level of a Junior-Senior undergraduate who has completed courses in Linear Algebra, Modern/Abstract Algebra, then this text is worth its weight in gold. For those individuals who have either chosen to make Mathematics their career or those who are Mathematically gifted, a text of this stature must be appreciated for exactly those reasons I used to 'negatively' criticize this text. For example, when doing research at any level above that of advanced undergraduate, the researcher should have the confidence, temperance, skill and desire to fill in missing gaps within proofs since the ability to do so is an excellent gauge of how well one actually understands the given material. It would seem to logically follow from this that the researcher would then benefit from choosing a text that contained exercises, which were not trivial calculations or the requirement of proving somthing that is either routine or standard. Instead, major rewards, in the form of confidence and a deeper understanding, are a result of struggling through difficult problems and, in general, problems which lead you toward self-discovery, i.e. those which are extensions of the given material. For these reasons I highly recommend this text to all members of the Mathematical community who desire more bang for their buck since this will serve them well, both as a text for further study and as a lifelong reference.