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

Serge Lang (Author)

Book Description

Publication Date: 1971 | ISBN-10: 0201041774 | ISBN-13: 978-0201041774 | Edition: Rev. print

This basic text for a one-year course in algebra at the graduate level thoroughly prepares students to handle the algebra they will use in all of mathematics. The author assumes that students have a basic familiarity with the language of mathematics "i.e.: sets and mapping, integers, and rational numbers." The text was thoroughly revised and enhanced in response to reviewers' comments and suggestions. Designed to improve students' retention and comprehension, the text is divided into four parts. The first introduces the basic notions of algebra. The second covers the direction of algebraic equations, including the Galois theory, and the final two parts cover the direction of linear and multilinear algebra.

--This text refers to an alternate Hardcover edition.

Editorial Reviews

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) --This text refers to an alternate Hardcover edition.

From the Back Cover

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. Comments on Serge Lang's 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. April 1999 Notices of the AMS, announcing that the author was awarded the Leroy P. Steele Prize for Mathematical Exposition for his many mathematics books. 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. MathSciNet's review of the first edition --This text refers to an alternate Hardcover edition.

Product Details

• Hardcover: 526 pages
• Publisher: Addison Wesley; Rev. print edition (1971)
• Language: German
• ISBN-10: 0201041774
• ISBN-13: 978-0201041774
• Average Customer Review: 4.0 out of 5 stars  See all reviews (24 customer reviews)
• Amazon Best Sellers Rank: #3,467,685 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

58 of 62 people found the following review helpful:

4.0 out of 5 stars a useful advanced graduate reference on algebra, April 4, 2006

By 

mathwonk - See all my reviews

This review is from: Algebra (Hardcover)

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.

43 of 47 people found the following review helpful:

4.0 out of 5 stars Excellent if you have the requisite mathematical maturity, December 26, 1998

By 

Timothy Chow - See all my reviews
(REAL NAME)   

This review is from: Algebra (Hardcover)

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.

32 of 36 people found the following review helpful:

5.0 out of 5 stars This will teach you how to run if you know how to walk, November 12, 2003

By A Customer

This review is from: Algebra (Hardcover)

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.