**Graduate Studies in Mathematics**

Volume: 90;
2008;
255 pp;
Hardcover

MSC: Primary 11; 12; 15;

Print ISBN: 978-0-8218-4465-6

Product Code: GSM/90

List Price: $60.00

Individual Member Price: $48.00

**Electronic ISBN: 978-1-4704-1159-6
Product Code: GSM/90.E**

List Price: $60.00

Individual Member Price: $48.00

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#### Supplemental Materials

# Basic Quadratic Forms

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*Larry J. Gerstein*

The arithmetic theory of quadratic forms is a rich branch of number theory that has had important applications to several areas of pure mathematics—particularly group theory and topology—as well as to cryptography and coding theory. This book is a self-contained introduction to quadratic forms that is based on graduate courses the author has taught many times. It leads the reader from foundation material up to topics of current research interest—with special attention to the theory over the integers and over polynomial rings in one variable over a field—and requires only a basic background in linear and abstract algebra as a prerequisite. Whenever possible, concrete constructions are chosen over more abstract arguments. The book includes many exercises and explicit examples, and it is appropriate as a textbook for graduate courses or for independent study. To facilitate further study, a guide to the extensive literature on quadratic forms is provided.

#### Table of Contents

# Table of Contents

## Basic Quadratic Forms

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface xi12 free
- Chapter 1. A Brief Classical Introduction 116 free
- Chapter 2. Quadratic Spaces and Lattices 1328
- §2.1. Fundamental Definitions 1328
- §2.2. Orthogonal Splitting; Examples of Isometry and Non-isometry 1631
- §2.3. Representation, Splitting, and Isotropy; Invariants u(F) and s(F) 2136
- §2.4. The Orthogonal Group of a Space 2641
- §2.5. Witt's Cancellation Theorem and Its Consequences 2944
- §2.6. Witt's Chain Equivalence Theorem 3449
- §2.7. Tensor Products of Quadratic Spaces; the Witt ring of a field 3550
- §2.8. Quadratic Spaces over Finite Fields 4055
- §2.9. Hermitian Spaces 4459

- Chapter 3. Valuations, Local Fields, and p-adic Numbers 5166
- §3.1. Introduction to Valuations 5166
- §3.3. Completions, Q[sub(p)], Residue Class Fields 5974
- §3.4. Discrete Valuations 6378
- §3.5. The Canonical Power Series Representation 6479
- §3.6. Hensel's Lemma, the Local Square Theorem, and Local Fields 6984
- §3.7. The Legendre Symbol; Recognizing Squares in Q[sub(p)] 7489

- Chapter 4. Quadratic Spaces over Q[sub(p)] 8196
- Chapter 5. Quadratic Spaces over Q 97112
- Chapter 6. Lattices over Principal Ideal Domains 113128
- §6.1. Lattice Basics 114129
- §6.2. Valuations and Fractional Ideals 116131
- §6.3. Invariant factors 118133
- §6.4. Lattices on Quadratic Spaces 122137
- §6.5. Orthogonal Splitting and Triple Diagonalization 124139
- §6.6. The Dual of a Lattice 128143
- §6.7. Modular Lattices 133148
- §6.8. Maximal Lattices 136151
- §6.9. Unimodular Lattices and Pythagorean Triples 138153
- §6.10. Remarks on Lattices over More General Rings 141156

- Chapter 7. Initial Integral Results 145160
- Chapter 8. Local Classification of Lattices 161176
- Chapter 9. The Local-Global Approach to Lattices 175190
- §9.1. Localization 176191
- §9.2. The Genus 178193
- §9.3. Maximal Lattices and the Cassels–Pfister Theorem 181196
- §9.4. Sums of Integer Squares (Second Version) 184199
- §9.5. Indefinite Unimodular Z-Lattices 188203
- §9.6. The Eichler-Kneser Theorem; the Lattice Z[sup(n)] 191206
- §9.7. Growth of Class Numbers with Rank 196211
- §9.8. Introduction to Neighbor Lattices 201216

- Chapter 10. Lattices over F[sub(q)][x] 207222
- Chapter 11. Applications to Cryptography 225240
- Appendix: Further Reading 241256
- Bibliography 245260
- Index 251266 free
- Back Cover Back Cover1274

#### Readership

Graduate students interested in number theory and algebra. Mathematicians seeking an introduction to the study of quadratic forms on lattices over the integers and related rings.

#### Reviews

*Basic Quadratic Forms
* is a great introduction to the theory of quadratic forms. The author is
clearly an expert on the area as well as a masterful teacher. ... It should be
included in the collection of any quadratic forms enthusiast.

-- MAA Reviews

Gerstein's book contains a significant amount of material that has not appeared anywhere else in book form. ... It is written in an engaging style, and the author has struck a good balance, presenting enough proofs and arguments to give the flavor of the subject without getting bogged down in too many technical details. It can be expected to whet the appetites of many readers to delve more deeply into this beautiful classical subject and its contemporary applications.

-- Andrew G. Earnest for Zentralblatt MATH