- Series: Undergraduate Texts in Mathematics
- Hardcover: 556 pages
- Publisher: Springer; 2nd edition (February 24, 2006)
- Language: English
- ISBN-10: 0387946802
- ISBN-13: 978-0387946801
- Product Dimensions: 6.1 x 1.2 x 9.2 inches
- Shipping Weight: 2 pounds (View shipping rates and policies)
- Average Customer Review: 10 customer reviews
- Amazon Best Sellers Rank: #1,862,524 in Books (See Top 100 in Books)
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Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Undergraduate Texts in Mathematics) 2nd Edition
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"I consider the book to be wonderful...The exposition is very clear, there are many helpful pictures, and there are a great many instructive exercises, some quite challenging...offers the heart and soul of modern commutative and algebraic geometry." -The American Mathematical Monthly
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This text is very clear and well-written, has tons of examples and develops geometric intuition for the subject quite well. Overall I think it is a great text for its target audience of undergraduate students with basically no background in abstract algebra. Unfortunately for a student who has had a course in abstract algebra the text goes too slowly, develops a lot of concepts they should already know in it and the explanations can be rather long at times.
I highly recommend this text for its intended audience or as a supplement to a more advanced text for a reader who wants more examples, clearer statements, a more thorough development or a good source for geometric intuition about the subject.
You need some algebra background although the book inserts some quick algebra in there. Example it explains what rings are, fields, ideals, etc. Basic ring theory. So dont expect anything too fancy.
After that, the book introduces the topic with every now and then backtracks to algebra for review. Example, it presents ideal properties, radical ideal properties before it introduces a definition from alg geo that includes them in there.
Essesntial, you work with fields, and polynomial rings, and affine space. So expect all the proofs to be focused on that, even though they may be true for any ring, not just polynomial rings. True for some not all.
Other than that, it is a good read, nothing fancy, pretty concrete and for self learning its pretty good. Check it out.
Just to echo prior reviews, the writing is clear, the pace perfect, and the voice is refreshingly non-condescending. There are so few math books which fit this bill I had to register my immense appreciation!
The book is robust and comprehensive enough for self-study. It errs on the side of explaining surrounding ring theory as you go which obviates the need for wiki-ing every two minutes. In addition, the computational approach (through copious examples and exercises) gives you ready and frequent sanity checks on your understanding.
Even if you simply want a foundation in "pure" algebraic geometry, this book will more than suffice. It does a remarkable job of motivating the journey into the more pure stuff, which even many pure books wouldn't sully themselves with. If an equality or implication only goes one way, they'll reliably provide counterexamples to demonstrate why. Hey, you may even find the detours into computers and robotics interesting (as did I)! As a follow-on text for a pure approach, try Perrin's book.
Finally, note that the second part of Madhu Sudan's Algebra and Computation course at MIT follows this book closely. So at least one top expert in the field seems to have a good opinion of the book.