A Course in Algebra by E. B. Vinberg (PDF)

Description

This is a comprehensive textbook on modern algebra written by an internationally renowned specialist. It covers material traditionally found in advanced undergraduate and basic graduate courses and presents it in a lucid style. The author includes almost no technically difficult proofs, and reflecting his point of view on mathematics, he tries wherever possible to replace calculations and difficult deductions with conceptual proofs and to associate geometric images to algebraic objects. The effort spent on the part of students in absorbing these ideas will pay off when they turn to solving problems outside of this textbook. Another important feature is the presentation of most topics on several levels, allowing students to move smoothly from initial acquaintance with the subject to thorough study and a deeper understanding. Basic topics are included, such as algebraic structures, linear algebra, polynomials, and groups, as well as more advanced topics, such as affine and projective spaces, tensor algebra, Galois theory, Lie groups, and associative algebras and their representations. Some applications of linear algebra and group theory to physics are discussed. The book is written with extreme care and contains over 200 exercises and 70 figures. It is ideal as a textbook and also suitable for independent study for advanced undergraduates and graduate students.

User’s Reviews

Editorial Reviews: Review “This is a masterly textbook on basic algebra. It is, at the same time, demanding and down-to-earth, challenging and user-friendly, abstract and concrete, concise and comprehensible, and above all extremely educating, inspiring and enlightening.” — –Zentralblatt MATH”Great book! The author’s teaching experience shows in every chapter.” — –E. Zelmanov, University of California, San Diego”Vinberg has written an algebra book that is excellent, both as a classroom text or for self-study. It starts with the most basic concepts and builds in orderly fashion to moderately advanced topics … Well motivated examples help the student … to master the material thoroughly, and exercises test one’s growing skill in addition to covering useful auxiliary facts … years of teaching abstract algebra have enabled Vinberg to say the right thing at the right time.” — –Irving Kaplansky, MSRI

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I think this is the best book on algebra one can find. The selection and organization of topics is impeccable, while the arguments of various theorems are clear and penetrating. I hope that more books of this quality will be published.

⭐Wonderful book! I recommend .

⭐This book is not to be dismissed quickly and lightly as a re-hash of many topics covered in the all too numerous algebra texts already available. Why? For at least 3 reasons, it should be more carefully considered. What are these reasons?1) It is published by the American Mathematical Society. (This is an indicator, not always reliable, of high quality.)2.) It is written by professor from Moscow State University, and mathematicians from this school and the Russian school of mathematics can be superb expositors.3.) It has an endorsement from Irving Kaplansky, one of the great American mathematicians of our times, on the back cover.On looking through the book I found it to be a very pleasant introduction to algebra. It is comprehensive, carefully written, and FRIENDLY. What an important thing for an introductory text to be! True, there are places where there are infelicities in translation, not many, and this is not to be unexpected. And there is the occasional spot where the author remarks on something that is SUPPOSED to seem obvious, at least the way he remarks on it. But it doesn’t seem obvious to me. So this can be a good thing, making the reader pause, pick up the pencil and paper, and start to work.I really like Chapter 7 on affine and projective spaces. And this is followed by Chapter 8 on tensor algebra. Vinberg points out that this latter subject is a language rather than a substantial theory, but a useful and in fact indispensable language that helps to unify all the objects of linear algebra. This kind of insight can only come from a master.The problems are selected well, and the book can be read on several levels. I conclude that it is written by a master, and should not be quickly dismissed. It is worth a look if you are looking for a masterly and friendly introduction to algebra, that great intermediary between number and geometry.

⭐This, in my opinion is a terrible book for a year sequence in graduate algebra. First and foremost, the exposition is terrible. He frequently leaves out non-trivial details which makes the text hard to work through. Also, does not spend enough time working through the complexities of group, field or ring theory. Those are almost an after thought to him. Although I appreciate his examples that show that algebra is not a self-contained field, that is more for a topics class not a two semester graduate sequence. Also, the index, which is one of the most important features of a graduate text book, is terrible.

⭐Two complaints.One major, one minor.The minor one is that the index is terrible to the point to being unusable. Literally nothing is listed in its logical place.The major one is that this is not a Graduate text. It’s comprehensive in the shallow, walking-along-the-coastline kind of way. Nothing is described in depth, which is fine for topics that don’t have any (like, say, linear algebra), but not fine for Galois theory (the section on which is atrocious) or Representation theory (which isn’t covered very well, even for finite groups).The strength of this book is that it covers many topic and in a nonstandard, more natural (to me anyway) way. That’s about it.

⭐The 2-star review is unfair. This book is very well written, rigorous and thoughtful. It covers an impressive range of topics, many of them are relevant to physics, the examples are many and usually clearly explained. The exercises interspersed throughout the text are just the right level of difficulty (most of them can be solved without having to bang your head against the wall). It is an excellent text for those wanting a good prelude to learning differential geometry. The last chapter on lie groups and algebras, the chapter on grassman and tensor algebra are good intros. It’s probably not as good as Artin’s text, but beats many of the other texts out there, and certainly more friendly to read yet covers advanced topics with all the rigor required.

⭐An excellent and clear boom ad reference for studying abstract and linear algebra

⭐This is one of the best written books for Algebra. Starting with introductory material and going on to more advanced material. It’s great for building up concepts and the author has a very clear and lucid style – definitions, propositions, proofs, exercises and explanatory material all build up the insight and basic concepts.

⭐There is no doubt that book is good but this is regarding service of Amezon. I found some half printed pages and some are totally blank. Pictures are attached with this review.

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