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Galois Theory: Lectures Delivered at the University of Notre Dame by Emil Artin (Notre Dame Mathematical Lectures, Number 2) Paperback – July 10, 1997

4.1 out of 5 stars 17 customer reviews

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Product Details

  • Paperback: 96 pages
  • Publisher: Dover Publications; 0002-Revised edition (July 10, 1997)
  • Language: English
  • ISBN-10: 0486623424
  • ISBN-13: 978-0486623429
  • Product Dimensions: 5.5 x 0.2 x 8.5 inches
  • Shipping Weight: 4.2 ounces (View shipping rates and policies)
  • Average Customer Review: 4.1 out of 5 stars  See all reviews (17 customer reviews)
  • Amazon Best Sellers Rank: #213,761 in Books (See Top 100 in Books)

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Format: Paperback
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.
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Format: Paperback
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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Format: Paperback
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.
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Format: Paperback Verified Purchase
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.
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Format: Paperback Verified Purchase
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.
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Format: Kindle Edition Verified Purchase
I expected that lectures on a theory might start out by giving a definition of that theory, or at least some indication of what the lectures would cover. But this book seems never to get to that point. Instead, (for example) the first section is devoted to linear algebra and the second paragraph tells us ". . . a field is a set of elements which. . .(form) an additive Abelian group . . ." In other words, it starts with the assumption that we are already familiar with group theory (but that we are unfamiliar with fields!). I have read it twice now and I still have no idea what the author's intent was or who was in his target audience.
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Format: Paperback
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!
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Format: Paperback Verified Purchase
This is a cute little book that guides the reader into Galois theory starting all the way from the review of linear algebra and polynomial rings over fields and progressing all the way to the Fundamental theorem. There are moreover many nice sections on Finite fields, Noether equations, Kummer extensions and as a final chapter the application to solvability by radicals of a general polynomial and the ruler and compass constructions. So the book is pretty self-contained and contains lots of good stuff. Also, Artin has a knack of giving very down-to-earth proofs that could be characterized as computational (rather than conceptual). It depends on everyone's preference whether they like this approach but for me it was very refreshing change of pace (compared to abstract and ofter almost magical proofs e.g. from commutative algebra).
In any case, patient reader will walk away from this book with a feeling of having built the subject from the ground up.

Nevertheless, I can't give it 5 stars because the book is very lacking in exercises. There are some applications scattered here and there (e.g. on symmetric extensions of function fields and on symmetric functions) but this is hopelessly insufficient to solidify the knowledge gained from the theorems. To properly understand Galois theory one needs to get their hands dirty by investigating splitting fields and Galois groups of all kinds of polynomials and paying close attention to the interaction of roots and group actions. In this regard the book leaves the reader completely on their own and so should be complemented by some additional source of exercices.
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