Hardcover ISBN:  9780821843291 
Product Code:  COLL/56 
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MAA Member Price:  $82.80 
AMS Member Price:  $73.60 
Electronic ISBN:  9781470432027 
Product Code:  COLL/56.E 
List Price:  $86.00 
MAA Member Price:  $77.40 
AMS Member Price:  $68.80 

Book DetailsColloquium PublicationsVolume: 56; 2008; 435 ppMSC: 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.ReadershipGraduate 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


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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.
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