作为作者获奖书Algebraic Theory of Quadratic Forms (Benjamin, 1973) 的新版，本书给出了在特征非 2 的任意域上的二次型理论的一个现代、自足的导引。从除了线性代数外的少量预备知识出发，作者讲述了一个专家级的课程，内容从二次型的Witt经典理论、四元数与Clifford 代数、形式实域的 Artin-Schreier理论、Witt 环的结构定理，到 Pfister形式理论、函数域和域不变量。这些主要进展与所涉及的 Brauer-Wall 群、局部与整体域、迹形式、Galois理论以及初等代数 K-理论天衣无缝地交织在一起，对域上二次型理论做了一个独一无二的原创性处理。新版中增加了超过100页全新的两章，内容包括这个领域中更新的结果以及更加近代的观点。 作为作者写作的特点，本书主要内容的陈述总是穿插大量精心挑选的解释一般理论的例题。这个特点再加上全部十三章280多个内容丰富的习题，极大提升了本书的价值，使得本书可以作为代数、数论、代数几何、代数拓扑以及几何拓扑研究者的参考书。
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
4. Scharlau's Norm Principle
5. Knebusch's Norm Principle
6. Galois Extensions and Trace Forms
7. Quadratic Closures of Fields
Exercises for Chapter VII
Chapter VIII. Formally Real Fields, Real-Closed Fields, and Pythagorean Fields
1. Structure of Formally Real Fields
2. Characterizations of Real-Closed Fields
Appendix A: Uniqueness of Real-Closure
Appendix B: Another Artin-Schreier Theorem
3. Pfister's Local-Global Principle
4. Pythagorean Fields
Appendix: Fields with 8 Square Classes and 20rderings
5. Connections with Galois Theory
6. Harrison Topology on XF
7. Prime Spectrum of W(F)
8. Applications to the Structure of W(F)
9. An Introduction to Preorderings
Exercises for Chapter VIII
Chapter IX. Quadratic Forms under Transcendental Extensions
1. Cassels-Pfister Theorem
2. Second and Third Representation Theorems
3. Milnor's Exact Sequence for W(F(x))
4. Scharlau's Reciprocity Formula for F(x)
Exercises for Chapter IX
Chapter X. Pfister Forms and Function Fields
1. Chain P-Equivalence
Appendix: Round Forms
2. Multiplicative Forms
3. Introduction to Function Fields
4. Basic Theorems on Function Fields
5. Hanptsatz, Linkage, and Forms in InF
6. Milnor's Higher K-Groups Exercises for Chapter X
Chapter XI. Field Invariants
1. Sums of Squares
2. The Level of a Field
3. Pfister-Witt Annihilator Theorem
4. The Property (An)
5. Height and Pythagoras Number
6. The u-Invariant of a Field
Appendix: The General u-Invariant
7. The Size of W(F), and C-Fields
Exercises for Chapter XI
Chapter XII. Special Topics in Quadratic Forms
1. Isomorphisms of Witt Rings
2. Quadratic Forms of Low Dimension
Appendix: Forms with Isomorphic Function Fields
3. Some Classification Theorems
4. Witt Rings under Biquadratic Extensions
5. Nonreal Fields with Eight Square Classes
6. Kaplansky Radical and Hilbert Fields
7. Construction of Some Pre-Hilbert Fields
8. Axiomatic Schemes for Quadratic Forms
Exercises for Chapter XII
Chapter XIII. Special Topics on Invariants
1. The u-Invariant of C((x, y))
2. Fields of u-Invariant 6
3. Fields of Pythagoras Number 6 and 7
4. Levels of Commutative Rings
5. Pythagoras Numbers of Commutative Rings
6. Some Open Questions
Exercises for Chapter XIII
Bibliography
Index
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