Algebra

Now I realize that my purchase of Lang's text was probably premature. It is a difficult text, and although I enjoy it, getting through any given chapter takes me a long time. I have not completed the text (as a matter of fact I am still in section I), but I feel that I can see the basic style of the text. Lang's text is well written (although I keep running into small errors), and probably should be in the library of any graduate student in the pure mathematics. As for me, I think I jumped at this book too soon and should go back to Artin and complete that text first. Lang's text is probably better suited for a second class in algebra at the graduate level (there are some other advanced tests such as Jacobson's, but I believe that Lang's text is the only well written text book in algebra which covers such a wide range of topics in proper detail).

In fall 1999, my freshman year, I will take graduate algebra and graduate real analysis, I hope after those classes I will be able to tackle Lang's text in a more reasonable manner.


a useful advanced graduate refernce on algebra   April 4, 2006
mathwonk
27 out of 30 found this review helpful

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.


Excellent if you have the requisite mathematical maturity   December 26, 1998
Timothy Chow
33 out of 37 found this review helpful

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.


Almost perfect!   January 18, 2006
James Curry (Sacramento, CA USA)
5 out of 7 found this review helpful

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?


Good one volume expostion of basic algebra   May 20, 1999
11 out of 12 found this review helpful

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.