Product Description

"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.

Product Details

• Amazon Sales Rank: #59174 in Books
• Published on: 2005-06-21
• Original language: English
• Number of items: 1
• Binding: Hardcover
• 912 pages

Review

S. Lang

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

From the reviews of the third edition:

"The current third edition has grown again … dealing with topics close to the author’s heart from number theory, function theory and algebraic geometry. For the math graduate who wants to broaden his education this is an excellent account; apart from standard topics it picks out many items from other fields … . This makes it a fascinating book to read … . a very readable treatment of many modern mainline topics as well as some interesting out-of-the-way items." (Paul M. Cohn, Zentralblatt MATH, Vol. 984, 2003)

"Lang’s Algebra … has gained an iconic status, due both to the comprehensiveness of its coverage and its ability to be authoritative and lively at the same time. … a revolutionary work, changing the way in which graduate algebra was taught. … the author describes the book as ‘very stable’, indicating that there is little that he has wished to change. This confidence is reflected in the wider mathematical community, and ... this new printing deserves a place in every university departmental library." (Gerry Leversha, The Mathematical Gazette, Vol. 87 (509), 2003)

Customer Reviews

A worthwhile pain in the [behind]
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.

Excellent if you have the requisite mathematical maturity
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.

a useful advanced graduate reference on algebra
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.