Customer Reviews

Galois Theory: Lectures Delivered at the University of Notre Dame by Emil Artin (Notre Dame Mathematical Lectures, Number 2)

5 star:		(6)
4 star:		(2)
3 star:		(2)
2 star:		(0)
1 star:		(1)

 

 

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.

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.

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.

Any student (graduate or undergraduate) who is learning Galois theory will benefit greatly from reading this book. Artin has a very elegant style of writing and many parts of the book read like a novel. At its current price, there's no reason to not buy this book; you may actually want to buy a few extra copies as they make great gifts and/or stocking stuffers.

I would also recommend Artin's Geometric Algebra.

Galois Theory is in traditional mathematical format. The major elements of the book are definitions, lemmas, theorems, and proofs. The book introduces the major topics of Galois Theory. They are fields, extension fields, splitting fields, unique decomposition of polynomials into irreducible factors, solvable groups, permutation groups, and solution of equations by radical.

The last part of the book contains the major results of Galois Theory with proofs using the theorems from the second part of the book. They are theorem 5: The polynomial f(x) is solvable by radicals if and only if its group is solvable; theorem 4: The symmetric group G on n letters is not solvable for n > 4; theorem 6: The group of the general equation of degree n is the symmetric group on n letters. The general equation of degree n is not solvable by radicals if n > 4.

This is my second Galois Theory book. What impress me most is the involvement to prove the major results of Galois Theory such as theorem 5 and theorem 6. In order to prove the theorems, mathematicians invent many mathematical objects. They are root, group, symmetric group, solvable group, field, extension field, splitting field, Kummer field/extension, Abelian group, normal subgroup, normal extension, factor/quotient group, homomorph, fixed field, extension by radicals field, and more. Nowadays, we put all these objects under the domain of abstract algebra.

The book is certainly not self-contained because one would need an abstract algebra textbook for reference to the mathematical objects.

In my opinion this is without a doubt the best text on the subject. Do not be fooled by the small size of this book. It is lean, to the point and refreshingly laconic. A masterpiece!

I agree, to some extent, with the recent two reviewers: Nobody can deny that Emil Artin was a great mathematician, having done a very good job in algebra. That does not necessarily mean his textbooks should be praised *ad infinitum*. I understand some classics remain valuable for an incredibly long period of time ("Morse theory" by Milnor is one of such landmarks that comes into my mind), but I feel scheptical if this one deserves that claim. This book is okay if you are interested in his writing style of many years ago, but not quite so if your main concern is to study Galois theory (or algebra: that makes no difference for that matter) efficiently and effectively. In that case you should turn to more modern textbooks like Cohn ("Algebra" published by Wiley.)

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.

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.

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.