Contents

Preface xi

Chapter 1. A Brief Classical Introduction 1

§1.1. Quadratic Forms as Polynomials 1

§1.2. Representation and Equivalence; Matrix Connections;

Discriminants 4

Exercises 7

§1.3. A Brief Historical Sketch, and Some References to the

Literature 7

Chapter 2. Quadratic Spaces and Lattices 13

§2.1. Fundamental Definitions 13

§2.2. Orthogonal Splitting; Examples of Isometry and Non-isometry 16

Exercises 20

§2.3. Representation, Splitting, and Isotropy; Invariants u(F) and

s(F) 21

§2.4. The Orthogonal Group of a Space 26

§2.5. Witt's Cancellation Theorem and Its Consequences 29

§2.6. Witt's Chain Equivalence Theorem 34

§2.7. Tensor Products of Quadratic Spaces; the Witt ring of a field 35

Exercises 39

§2.8. Quadratic Spaces over Finite Fields 40

§2.9. Hermitian Spaces 44

Exercises 49

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