**Colloquium Publications**

Volume: 56;
2008;
435 pp;
Hardcover

MSC: Primary 11; 14;

**Print ISBN: 978-0-8218-4329-1
Product Code: COLL/56**

List Price: $92.00

AMS Member Price: $73.60

MAA Member Price: $82.80

**Electronic ISBN: 978-1-4704-3202-7
Product Code: COLL/56.E**

List Price: $86.00

AMS Member Price: $68.80

MAA Member Price: $77.40

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#### Supplemental Materials

# The Algebraic and Geometric Theory of Quadratic Forms

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*Richard Elman; Nikita Karpenko; Alexander Merkurjev*

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.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## The Algebraic and Geometric Theory of Quadratic Forms

- Cover Cover11
- Title page i2
- Dedication iii4
- Contents v6
- Introduction 110
- Classical theory of symmetric bilinear forms and quadratic forms 918
- Bilinear forms 1120
- Quadratic forms 3948
- Forms over rational function fields 7180
- Function fields of quadrics 93102
- Bilinear and quadratic forms and algebraic extensions 121130
- 𝑢-invariants 161170
- Applications of the Milnor conjecture 177186
- On the norm residue homomorphism of degree two 193202
- Algebraic cycles 215224
- Homology and cohomology 217226
- Chow groups 261270
- Steenrod operations 277286
- Category of Chow motives 291300
- Quadratic forms and algebraic cycles 305314
- Cycles on powers of quadrics 307316
- The Izhboldin dimension 325334
- Application of Steenrod operations 335344
- The variety of maximal totally isotropic subspaces 355364
- Motives of quadrics 371380
- Appendices 381390
- Bibliography 421430
- Notation 427436
- Terminology 431440
- Back Cover Back Cover1448