**Graduate Studies in Mathematics**

Volume: 67;
2005;
550 pp;
Hardcover

MSC: Primary 11;
Secondary 15

Print ISBN: 978-0-8218-1095-8

Product Code: GSM/67

List Price: $89.00

Individual Member Price: $71.20

**Electronic ISBN: 978-1-4704-2108-3
Product Code: GSM/67.E**

List Price: $89.00

Individual Member Price: $71.20

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#### Supplemental Materials

# Introduction to Quadratic Forms over Fields

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*T.Y. Lam*

This new version of the author's prizewinning book, *Algebraic Theory of
Quadratic Forms* (W. A. Benjamin, Inc., 1973), gives a modern and
self-contained introduction to the theory of quadratic forms over fields of
characteristic different from two. Starting with few prerequisites beyond
linear algebra, the author charts an expert course from Witt's classical theory
of quadratic forms, quaternion and Clifford algebras, Artin-Schreier theory of
formally real fields, and structural theorems on Witt rings, to the theory of
Pfister forms, function fields, and field invariants. These main developments
are seamlessly interwoven with excursions into Brauer-Wall groups, local and
global fields, trace forms, Galois theory, and elementary algebraic K-theory,
to create a uniquely original treatment of quadratic form theory over fields.
Two new chapters totaling more than 100 pages have been added to the earlier
incarnation of this book to take into account some of the newer results and
more recent viewpoints in the area.

As is characteristic of this author's expository style, the presentation of the main material in this book is interspersed with a copious number of carefully chosen examples to illustrate the general theory. This feature, together with a rich stock of some 280 exercises for the thirteen chapters, greatly enhances the pedagogical value of this book, both as a graduate text and as a reference work for researchers in algebra, number theory, algebraic geometry, algebraic topology, and geometric topology.

#### Table of Contents

# Table of Contents

## Introduction to Quadratic Forms over Fields

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface xii13 free
- Notes to the Reader xvii18 free
- Partial List of Notations xix20 free
- Chapter I. Foundations 124 free
- §1. Quadratic Forms and Quadratic Spaces 124
- §2. Diagonalization of Quadratic Forms 528
- §3. Hyperbolic Plane and Hyperbolic Spaces 932
- §4. Decomposition Theorem and Cancellation Theorem 1235
- §5. Witt's Chain Equivalence Theorem 1538
- §6. Kronecker Product of Quadratic Spaces 1740
- §7. Generation of the Orthogonal Group by Reflections 1841
- Exercises for Chapter I 2245

- Chapter II. Introduction to Witt Rings 2750
- Chapter III. Quaternion Algebras and their Norm Forms 5174
- Chapter IV. The Brauer-Wall Group 79102
- Chapter V. Clifford Algebras 103126
- §1. Construction of Clifford Algebras 103126
- §2. Structure Theorems 108131
- §3. The Clifford Invariant, Witt Invariant, and Hasse Invariant 113136
- §4. Real Periodicity and Clifford Modules 122145
- §5. Composition of Quadratic Forms 127150
- §6. Steinberg Symbols and Milnor's Group k[sub(2)]F 132155
- Exercises for Chapter V 140163

- Chapter VI. Local Fields and Global Fields 143166
- Chapter VII. Quadratic Forms Under Algebraic Extensions 187210
- §1. Scharlau's Transfer 187210
- §2. Simple Extensions and Springer's Theorem 191214
- §3. Quadratic Extensions 196219
- §4. Scharlau's Norm Principle 204227
- §5. Knebusch's Norm Principle 206229
- §6. Galois Extensions and Trace Forms 209232
- §7. Quadratic Closures of Fields 218241
- Exercises for Chapter VII 226249

- Chapter VIII. Formally Real Fields, Real-Closed Fields, and Pythagorean Fields 231254
- §1. Structure of Formally Real Fields 231254
- §2. Characterizations of Real-Closed Fields 240263
- §3. Pfister's Local-Global Principle 252275
- §4. Pythagorean Fields 255278
- §5. Connections with Galois Theory 267290
- §6. Harrison Topology on X[sub(f)] 271294
- §7. Prime Spectrum of W(F) 277300
- §8. Applications to the Structure of W(F) 281304
- §9. An Introduction to Preorderings 288311
- Exercises for Chapter VIII 292315

- Chapter IX. Quadratic Forms under Transcendental Extensions 299322
- Chapter X. Pfister Forms and Function Fields 315338
- Chapter XI. Field Invariants 375398
- Chapter XII. Special Topics in Quadratic Forms 425448
- §1. Isomorphisms of Witt Rings 426449
- §2. Quadratic Forms of Low Dimension 431454
- §3. Some Classification Theorems 439462
- §4. Witt Rings under Biquadratic Extensions 443466
- §5. Nonreal Fields with Eight Square Classes 447470
- §6. Kaplansky Radical and Hilbert Fields 450473
- §7. Construction of Some Pre-Hilbert Fields 456479
- §8. Axiomatic Schemes for Quadratic Forms 463486
- Exercises for Chapter XII 476499

- Chapter XIII. Special Topics on Invariants 479502
- Bibliography 533556
- Index 543566
- Back Cover Back Cover1577

#### Readership

Graduate students and research mathematicians interested in algebra, number theory, algebraic geometry, algebraic topology, and geometric topology.

#### Reviews

The book reads very well; notions and statements are supported by examples involving cases over both finite and infinite fields. At the end of every chapter, there are a number of exercises that could be useful, especially for teachers using the book as a basis for their course.

-- EMS Newsletter

(This) book is a wonderful achievement. Its genesis is recounted with charm and warmth in the preface. The author's lucid style and expository skill, his judicious choice of topics and their impeccable layout, not to mention the beautiful typesetting make the book a joy to read or just to browse. it will be a must for anybody working in quadratic forms or on topics related to or using quadratic forms, be it for learning the theory of quadratic forms over fields from its foundations, or be it as a reference.

-- Zentralblatt Math