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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