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(More customer reviews)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.
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AlgebraThis 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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