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The Algebraic and Geometric Theory of Quadratic Forms

Richard Elman University of California, Los Angeles, CA
Nikita Karpenko Université Pierre et Marie Curie - Paris 6, Paris, France
Alexander Merkurjev University of California, Los Angeles, CA
Available Formats:
Hardcover ISBN: 978-0-8218-4329-1
Product Code: COLL/56
435 pp
List Price: $92.00 MAA Member Price:$82.80
AMS Member Price: $73.60 Electronic ISBN: 978-1-4704-3202-7 Product Code: COLL/56.E 435 pp List Price:$86.00
MAA Member Price: $77.40 AMS Member Price:$68.80
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This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $138.00 MAA Member Price:$124.20
AMS Member Price: $110.40 Click above image for expanded view The Algebraic and Geometric Theory of Quadratic Forms Richard Elman University of California, Los Angeles, CA Nikita Karpenko Université Pierre et Marie Curie - Paris 6, Paris, France Alexander Merkurjev University of California, Los Angeles, CA Available Formats:  Hardcover ISBN: 978-0-8218-4329-1 Product Code: COLL/56 435 pp  List Price:$92.00 MAA Member Price: $82.80 AMS Member Price:$73.60
 Electronic ISBN: 978-1-4704-3202-7 Product Code: COLL/56.E 435 pp
 List Price: $86.00 MAA Member Price:$77.40 AMS Member Price: $68.80 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price:$138.00
MAA Member Price: $124.20 AMS Member Price:$110.40
• Book Details

Colloquium Publications
Volume: 562008
MSC: Primary 11; 14;

This book is a comprehensive study of the algebraic theory of quadratic forms, from classical theory to recent developments, including results and proofs that have never been published. The book is written from the viewpoint of algebraic geometry and includes the theory of quadratic forms over fields of characteristic two, with proofs that are characteristic independent whenever possible. For some results both classical and geometric proofs are given.

Part I includes classical algebraic theory of quadratic and bilinear forms and answers many questions that have been raised in the early stages of the development of the theory. Assuming only a basic course in algebraic geometry, Part II presents the necessary additional topics from algebraic geometry including the theory of Chow groups, Chow motives, and Steenrod operations. These topics are used in Part III to develop a modern geometric theory of quadratic forms.

Readership

Graduate students and research mathematicians interested in algebraic geometry and number theory.

• Table of Contents

• Chapters
• Introduction
• Classical theory of symmetric bilinear forms and quadratic forms
• Chapter 1. Bilinear forms
• Chapter 2. Quadratic forms
• Chapter 3. Forms over rational function fields
• Chapter 4. Function fields of quadrics
• Chapter 5. Bilinear and quadratic forms and algebraic extensions
• Chapter 6. $u$-invariants
• Chapter 7. Applications of the Milnor conjecture
• Chapter 8. On the norm residue homomorphism of degree two
• Algebraic cycles
• Chapter 9. Homology and cohomology
• Chapter 10. Chow groups
• Chapter 11. Steenrod operations
• Chapter 12. Category of Chow motives
• Quadratic forms and algebraic cycles
• Chapter 13. Cycles on powers of quadrics
• Chapter 14. The Izhboldin dimension
• Chapter 15. Application of Steenrod operations
• Chapter 16. The variety of maximal totally isotropic subspaces
• Chapter 17. Motives of quadrics
• Appendices
• Additional Material

• Reviews

• This book is a welcome addition to the vast literature on quadratic forms, summarizing recent advances of the theory, highlighting the algebraic geometry approach and shedding new light on the classical results.

MAA Reviews
• Overall, this book is an outstanding achievement and will be an indispensable reference for specialists, though challenging for beginners.

Mathematical Reviews
• The first part of the book may well serve as a modern introduction to the classical algebraic theory of quadratic forms ... the exposition is throughout lucid, detailed, enlightening and inspiring, very much to the benefit of the keen reader. No doubt, the authors have done an admirable and rewarding job by making such a modern, encyclopedic text available to the mathematical community as a whole.

Zentralblatt MATH
• Request Review Copy
• Get Permissions
Volume: 562008
MSC: Primary 11; 14;

This book is a comprehensive study of the algebraic theory of quadratic forms, from classical theory to recent developments, including results and proofs that have never been published. The book is written from the viewpoint of algebraic geometry and includes the theory of quadratic forms over fields of characteristic two, with proofs that are characteristic independent whenever possible. For some results both classical and geometric proofs are given.

Part I includes classical algebraic theory of quadratic and bilinear forms and answers many questions that have been raised in the early stages of the development of the theory. Assuming only a basic course in algebraic geometry, Part II presents the necessary additional topics from algebraic geometry including the theory of Chow groups, Chow motives, and Steenrod operations. These topics are used in Part III to develop a modern geometric theory of quadratic forms.

Readership

Graduate students and research mathematicians interested in algebraic geometry and number theory.

• Chapters
• Introduction
• Classical theory of symmetric bilinear forms and quadratic forms
• Chapter 1. Bilinear forms
• Chapter 2. Quadratic forms
• Chapter 3. Forms over rational function fields
• Chapter 4. Function fields of quadrics
• Chapter 5. Bilinear and quadratic forms and algebraic extensions
• Chapter 6. $u$-invariants
• Chapter 7. Applications of the Milnor conjecture
• Chapter 8. On the norm residue homomorphism of degree two
• Algebraic cycles
• Chapter 9. Homology and cohomology
• Chapter 10. Chow groups
• Chapter 11. Steenrod operations
• Chapter 12. Category of Chow motives
• Quadratic forms and algebraic cycles
• Chapter 13. Cycles on powers of quadrics
• Chapter 14. The Izhboldin dimension
• Chapter 15. Application of Steenrod operations
• Chapter 16. The variety of maximal totally isotropic subspaces
• Chapter 17. Motives of quadrics
• Appendices
• This book is a welcome addition to the vast literature on quadratic forms, summarizing recent advances of the theory, highlighting the algebraic geometry approach and shedding new light on the classical results.

MAA Reviews
• Overall, this book is an outstanding achievement and will be an indispensable reference for specialists, though challenging for beginners.

Mathematical Reviews
• The first part of the book may well serve as a modern introduction to the classical algebraic theory of quadratic forms ... the exposition is throughout lucid, detailed, enlightening and inspiring, very much to the benefit of the keen reader. No doubt, the authors have done an admirable and rewarding job by making such a modern, encyclopedic text available to the mathematical community as a whole.

Zentralblatt MATH
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