Preface
Notes to the Reader
Partial List of Notations
Chapter I. Foundations
1. Quadratic Forms and Quadratic Spaces
2. Diagonalization of Quadratic Forms
3. Hyperbolic Plane and Hyperbolic Spaces
4. Decomposition Theorem and Cancellation Theorem
5. Witt's Chain Equivalence Theorem
6. Kronecker Product of Quadratic Spaces
7. Generation of the Orthogonal Group by Reflections
Exercises for Chapter I
Chapter II. Introduction to Witt Rings
1. Definition of W(F) and W(F)
2. Group of Square Classes
3. Some Elementary Computations
4. Presentation of Witt Rings
5. Classification of Small Witt Rings
Exercises for Chapter II
Chapter III. Quaternion Algebras and their Norm Forms
1. Construction of Quaternion Algebras
2. Quaternion Algebras as Quadratic Spaces
3. Coverings of the Orthogonal Groups
4. Linkage of Quaternion Algebras
5. Characterizations of Quaternion Algebras Exercises for Chapter III
Chapter IV. The Brauer-Wall Group
1. The Brauer Group
2. Central Simple Graded Algebras (CSGA)
3. Structure Theory of CSGA
4. The Brauer-Wall Group Exercises for Chapter IV
Chapter V. Clifford Algebras
1. Construction of Clifford Algebras
2. Structure Theorems
3. The Clifford Invariant, Witt Invariant, and Hasse Invariant
4. Real Periodicity and Clifford Modules
5. Composition of Quadratic Forms
6. Steinberg Symbols and Milnor's Group k2F
Exercises for Chapter V
Chapter VI. Local Fields and Global Fields
1. Springer's Theorem for C.D.V. Fields
2. Quadratic Forms over Local Fields
Appendix: Nonreal Fields with Four Square Classes
3. Hasse-Minkowski Principle
4. Witt Ring of Q
5. Hilbert Reciprocity and Quadratic Reciprocity
Exercises for Chapter VI
Chapter VII. Quadratic Forms Under Algebraic Extensions
1. Scharlau's Transfer
2. Simple Extensions and Springer's Theorem
3. Quadratic Extensions
