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Rating: 
Summary: just enjoy
Review: A friend of mine has a maxim: The shorter a math book, the more likely I am to read it. Artin's Galois Theory is certainly that. It is also an example of Artin's wonderful mathematical style. Gian-Carlo Rota, who took classes from Artin when he was at princeton, said that Artin's proofs were perfect, as though he had gone through all the available proofs to find *the* proof and that was the one he used. Rota felt that this left the student at a disadvantage in that he didn't know about the effort that went into the proof, nor why it is beautiful. I disagree: the proofs in Galois Theory have a certain indescribable beauty to them which left me awestruck at their simplicity. They seem to have all the requisite attributes (as laid out in Hardy's A Mathematician's apology) to be considered beautiful. These notes are by no means complete, but I would suggest them as a suplement to another treatment of field theory (for example, Dummit and Foote or Morandi even though they were based upon Artin's treatment).
Rating:
Summary: just enjoy
Review: during reading this cute booklet, you can surely hear the gentle talk of an old math maven.(from the publishing date, the auther was 44 but that's my impression.) with a cup of coffee, stretch those edgy wrinkles of your brain.
Rating:
Summary: Succinct exposition of modern Galois theory by a pioneer.
Review: Emil Artin's short book gets a mention in most texts on
Galois theory. It is very short - only 60 odd pages. Yet
it is a very clear, complete and readable account of the
essential elements of modern Galois theory. It is based
on lectures he gave over 50 years ago but you might think
it was written only yesterday and is comprehensible to
anyone familiar with current abstract algebra terminology.
And the price makes it a bargain. There are no worked
examples, exercises or index here.
Rating: 
Summary: Maybe a classic but not worthwile for me
Review: Ever since I took intermediate algebra high school, I've wanted to learn the proof for the insolvability of the general quintic polynomial. I bought this book with great hope and expectation, but, with all due respect to the previous Amazon reviewers and the late professor Artin, I found it severely lacking. Its proofs are too dense in some places and too sparse in others, and its notation is obscure. I returned it almost immediately. I do not believe that any eighth grader could understand this book. If its price were not so low and its potential audience so limited, I would suspect fraud on the part of the first amazon reviewer. I eventually used the 2nd edition of Fraleigh's Abstract Algebra text to learn the proof for the insolvability of the general quintic. Fraleigh leaves key parts of the proof as exercises for the reader, but if you have the patience to prove some theorems yourself, Fraleigh is the way to go.
Rating: 
Summary: Okay if you are interested in matehmatical "classics".
Review: Most of my teachers in number theory and algebra recommend this book as being the standard treatment of the subject. The subject is not at all simple, it requires a non trivial amount of abstract algebra. Moreover, what makes the subject relatively more important is its being a component of Wile's proof of Fermat's Last Theorm. As you can guess from my rating the book was not so satisfactory for me. The edition that I read was I guess the first edition of the book, it did not have any index or preface, but that did not interfer in my gudgement. The thing I hated the most about this book, is that it uses old notation, which made the book wordy and less understandable. In the beginning of the book Dr Artin proves some results from linear algebra as if he assumes that people know nothing about it, but then later in the book he uses groups and quotient groups without defining them which implies that you should know something about group theory, but usually people with good knowledge in group theory are even more knowledgable in vector space theory. And I guess the moral here is that you really should have had some training in abstract algebra before you have read this book and for that I recommend Herstein's "abstract algebra" for those of you who have not had any course in group theory or Jacobson's "Basic Algebra I" Chapters I & II for advanced students. For a better a book on the subject (only for advanced students) I recommend Jacobson's "Basic Algebra I" Chapter IV.
Rating:
Summary: Great even if you're not an Eighth grader
Review: This book is one of the very best that Dover has out there. In my opinion, it is the ultimate book on Galois theory. All treatments written since this one were based on it, and do not add anything fundamentally new. There are only two things about this book which one could potentially complain about: 1) The awful cover. 2) There are no exercises because the book is just based on lecture notes. But that's forgivable, because there is no other exposition this good of Galois theory. One wonderful thing about this book is that it is entirely self-contained. It starts by proving the few basic results from linear algebra it needs, and then builds from there in a beautiful way until the fundamental theorems of Galois theory have been proven in a most transparent way. Then, in the appendix, not by Artin, a few results from group theory are proven, just enough for the classical applications to the solvability of the quintic. Every proof in this book is very clear and I cannot imagine how one could improve on any of them. ET Bell claimed in one of his books that anyone who knew high school algebra could easily understand Galois's proof of the unsolvability of the quintic. I didn't believe that until I saw this book, which proves that ET Bell was absolutely correct.
Rating: 
Summary: the source!
Review: This is modern Galois Theory, straight from the horse's mouth! Galois Theory is taught today using field extensions rather than by actually solving polynomials, students also learn to view a field extension as a vector space over the smaller field; both of these things were pioneered by Artin. The book also has short, clear proofs of all the main theorems. The only problem is that there are no problems to work on, so I have to say this is only a good reference for Galois Theory.
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