Customer Reviews

Algebra. Revised Printing (Addison-Wesley series in mathematics)

5 star:		(9)
4 star:		(8)
3 star:		(5)
2 star:		(1)
1 star:		(1)

 

 

As others have said, this is not a book to begin learning algebra, but is a necessary book for most students to have on their shelves. Why is that? Basic topics are discussed from scratch in this book from the most advanced possible viewpoint. Hence few can learn them here for the first time, but no one can graduate to professional status without eventually arriving at this perspective.

In particular the categorical point of view is simply essential to a research mathematician to acquire at some point, and Lang uses it here from the beginning, while Dummitt and Foote place it in appendix II, after page 800. So Lang's goal seems not to introduce basic algebra, but to provide essential algebraic facts not found elsewhere, and to give them all from a professional's perspective.

This is probably a third book on algebra in today's world, and that is assuming the student is pretty good. The only current book I know of out there that is really aimed at students and also written by a top professional is Artin. If you can, begin with Artin, then read Dummitt and Foote for topics Artin omits, then read Lang to see how you should view the same material and find things Dummitt and Foote left out.

Then you are ready to do research with these tools. For instance one of our research professors tells his students the prerecquisite for working in algebraic number theory is to become comfortable with algebra at the level of Lang. But our course in PhD prelim preparation for algebra will probably use Dummitt and Foote, just because it is a more feasible book for the students to read at that stage. Attempts to use Lang in trhe past have been disastrous.

Nonetheless, even students who found Lang a frustrating text, still use it as a necessary reference, and even find it has too little.

Just compare the treatment of groups in Lang and Dummitt and Foote. Lang covers the whole subject in more depth in 60 pages (2nd edition) while D/F use up over 220 pages on groups, and still do not introduce the categorical point of view, and in particular do not prove the existence of "direct sums" i.e. coproducts (which they do not even define), of groups.

So if you only have Lang, you will almost surely not see enough detail to understand the material, and if you only have D/F you will not see it from quite the right perspective, and will still not know some basic results.

Lang's book has numerous frustrating traits, misprints, errors, many uses of the word "obvious" for arguments that need a great deal of filling in, careless slipups ad nauseam, dyslexic things like saying clearly when to use product as opposed to coproduct, then getting it precisely backwards himself. or a whole discussion of Galois groups as permutations of roots of polynomials while forgetting to assume the polynomial is separable.

Your margins in Lang will be full of corrections, comments and added details, but now and then he will make something so clear in a word or two, that it will forever seem easy to you. In sum it is a locally flawed and carelessly written book, but globally impressive, and one for which there is no adequate substitute to my knowledge. Not least, Addison Wesley has always done a good job of making the type look beautiful on the page. The integrity of some recent bindings of course is another story.

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.

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.

After having struggled with this book for most of my first year algebra class I grew to hate it! I now find myself refering to this book time and time again. This book really has it all. If you already have a good background in algebra and want a great referance, this book is a must. If you are taking algebra for the first time I would avoid this book and look at another intro. to algebra text like Grove (one of my favorite math texts).

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.

Difficult to agree with my learned friend from Jackson,Mississippi that the chapters on groups and rings were boring. I must congratulate him on finishing the book in one week. More seriously, the book provides enough coverage of commutative algebra,Galois theory and homological algebra as to enable one to tackle the books by Eisenbud and Hartshorne on commutative algebra and algebraic geometry respectively.There are rival treatments by Cohn and Jacobsen but Lang beats them for conciseness.Lang is notorious for errors and omissions in his books and so one would expect a reader to have considerable 'maturity', i.e. the ability to correct proofs or fill in missing details.

I have owned this book for many years now and I find that even after I have learned a topic in Lang or elsewhere, I still find that yet another careful reading of Lang will rearrange my thinking. I hear and read a lot of people complaining about this book, and I think most criticisms come from not having enough patience or energy to climb the book. (Yes, reading this is more like climbing!) There are few better things that you can do for yourself than hanging out with Lang.

The book does deserve some criticisms. His chapter on groups is just too small and insubstantial. Go elsewhere for that, like Rotman. The real purpose of that chapter is to introduce Category Theory, and it takes the wrong tack a few times there, I feel. So learn category theory somewhere else too. And all algebra books fail to explain what the algebra is good for. This one is no different. It is a shame because too many people think that Algebra is mostly for algebraists. But the truth is you can't do anything great without algebra.

The chapters on Homology theory are good in places, and the places where they are not so good, try the book by Weibel.

So, yeah, he is often a bit terse and leaves steps out. That's just an invitation to think things over. And it keeps the text clean. He is respecting you, honoring you, inviting you to the real party. He's not cheating you. He's giving you the real goo! You want the real goo, don't you?

I hope that in the coming reprint the author would take care to correct the numerous typos and errors in the 3rd edition. Though books are seldom free from mistakes, those in this book are particularly frustrating and confusing. The author did a lot of rearrangements from the 2nd to the 3rd edition, but many self-references in the book are not updated, and I could never figure out where the author refers to at some places.

Otherwise, I agree with the general opinion that this book is good but often too terse.

This book is like an encyclopia -- very thorough, but also somewhat inaccessable to those who don't already know the language and terms it uses. For a one volume summary of basic algebra, this book probably has the widest degree of coverage.

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".