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Algebra (Graduate Texts in Mathematics) [Hardcover]

by Serge Lang
4.0 out of 5 stars  See all reviews (17 customer reviews)

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Book Description

September 19, 2005 038795385X 978-0387953854 3rd
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. This book successfully addresses all of the basic concepts of algebra. For the new edition, the author has added exercises and made numerous corrections to the text. From MathSciNet's review of the first edition: "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."

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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)

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

Product Details

  • Series: Graduate Texts in Mathematics (Book 211)
  • Hardcover: 917 pages
  • Publisher: Springer; 3rd edition (September 19, 2005)
  • Language: English
  • ISBN-10: 038795385X
  • ISBN-13: 978-0387953854
  • Product Dimensions: 9.3 x 6.5 x 2.3 inches
  • Shipping Weight: 3.2 pounds (View shipping rates and policies)
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (17 customer reviews)
  • Amazon Best Sellers Rank: #141,135 in Books (See Top 100 in Books)

Customer Reviews

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Most Helpful Customer Reviews
93 of 98 people found the following review helpful
4.0 out of 5 stars a useful advanced graduate reference on algebra April 4, 2006
Format: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.
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41 of 46 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
Format: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.
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17 of 18 people found the following review helpful
5.0 out of 5 stars There is a website dedicated to this book January 8, 2009
Format:Hardcover
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".
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15 of 17 people found the following review helpful
4.0 out of 5 stars Almost perfect! January 18, 2006
Format:Hardcover
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?
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Most Recent Customer Reviews
5.0 out of 5 stars My Impression of Lang's Algebra
My review will be brief, as others have given much more detailed and insightful reviews. I only hope to give an impressionistic expression of how I think about this book:... Read more
Published 24 months ago by Adam
3.0 out of 5 stars Not a good text, but probably very useful for algebraists
We're using this book for my graduate level algebra course, and I can't stand it as a text book. Lang managed to cram an incredible wealth of information in this volume (no... Read more
Published on September 11, 2011 by From Detroit
3.0 out of 5 stars Terrible first algebra book
After having it used as my undergraduate algebra text, I hated algebra for about 15 years. Absolutely NO motivation or "genetic" understanding of the material. Read more
Published on March 7, 2010 by Narada
5.0 out of 5 stars Comprehensive book, great for reference.
This book covers a very wide variety of topics in algebra, and has lots of exercises. It is a great book for reference.
Published on October 30, 2009 by Mark
4.0 out of 5 stars Indispensable reference, but not perfect
Lang's Algebra is a very broad reference in abstract algebra: most subfields are at least mentioned, and there is a wealth of detail on many important topics. Read more
Published on July 4, 2009 by Cloudy Skies
5.0 out of 5 stars Best Algebra Book.
This is among the finest references for Algebra I have ever seen. Comprehensive is an understatement. Read more
Published on March 10, 2009 by GaloisGuy
4.0 out of 5 stars A great reference book, not good to learn from
As most of the books written by Prof. Lang, this book is a great compilation, it has what it needs to have, nothing more, nothing less. Read more
Published on May 13, 2007 by David Jimenez
5.0 out of 5 stars The way to learn algebra
Most critisism of the book are based in the fact that topic are not treated in deep, and the reason is that there is no need to do that. Read more
Published on July 18, 2006 by Hector Luis Gimbatti
5.0 out of 5 stars This book grows on you.
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. Read more
Published on May 2, 2006 by Alexander C. Zorach
1.0 out of 5 stars As a textbook, this book is terrible
I studied Lang's algebra as a Ph.D. student at Stanford University during my first year there. I will certainly never use it to teach a course myself. Read more
Published on September 23, 2005 by Alan
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