●Introduction
Part 1.Classical theory of symmetric bilinear forms and quadratic forms
Chapter Ⅰ.Bilinear Forms
1.Foundations
2.The Witt and Witt-Grothendieck rings of symmetric bilinear forms
3.Chain equivalence
4.Structure of the Witt ring
5.The Stiefel-Whitney map
6.Bilinear Pfister forms
Chapter Ⅱ.Quadratic Forms
7.Foundations
8.Witt's Theorems
9.Quadratic Pfister forms Ⅰ
10.Totally singular forms
11.The Clifford algebra
12.Binary quadratic forms and quadratic algebras
13.The discriminant
14.The Clifford invariant
15.Chain p-equivalence of quadratic Pfister forms
16.Cohomological invariants
Chapter Ⅲ.Forms over Rational Function Fields
17.The Cassels-Pfister Theorem
18.Values of forms
19.Forms over a discrete valuation ring
20.Similarities of forms
21.An exact sequence for W(F(t))
Chapter Ⅳ.Function Fields of Quadrics
22.Quadrics
23.Quadratic Pfister forms Ⅱ
24.Linkage of quadratic forms
25.The submodule Jn(F)
26.The Separation Theorem
27.A further characterization of quadratic Pfister forms
28.Excellent quadratic forms
29.Excellent field extensions
30.Central simple algebras over function fields of quadratic forms
Chapter Ⅴ.Bilinear and Quadratic Forms and Algebraic Extensions
31.Structure of the Witt ring
32.Addendum on torsion
33.The total signature
34.Bilinear and quadratic forms under quadratic extensions
35.Torsion in In(F)and torsion Pfster forms
Chapter Ⅵ.u-invariants
36.The u-invariant
37.The u-invariant for formally real fields
38.Construction of fields with even u-invariant
39.Addendum: Linked fields and the Hasse number
Chapter Ⅶ.Applications of the Milnor Conjecture
40.Exact sequences for quadratic extensions
41.Annihilators of Pfister forms
42.Presentation of In(F)
43.Going down and torsion-freeness
Chapter Ⅷ.On the Norm Residue Homomorphism of Degree Two
44.The main theorem
45.Geometry of conic curves
46.Key exact sequence
47.Hilbert Theorem 90 for K2
48.Proof of the main theorem
Part 2.Algebraic cycles
Chapter Ⅸ.Homology and Cohomology
49.The complex C*(X)
50.External products
51.Deformation homomorphisms
52.K-homology groups
53.Euler classes and projective bundle theorem
54.Chern classes
55.Gysin and pull-back homomorphisms
56.K-cohomology ring of smooth schemes
Chapter Ⅹ.Chow Groups
57.Definition of Chow groups
58.Segre and Chern classes
Chapter Ⅺ.Steenrod Operations
59.Definition of the Steenrod operations
60.Properties of the Steenrod operations
61.Steenrod operations for smooth schemes
Chapter Ⅻ.Category of Chow Motives
62.Correspondences
63.Categories of correspondences
64.Category of Chow motives
65.Duality
66.Motives of cellular schemes
67.Nilpotence Theorem
Part 3.Quadratic forms and algebraic cycles
Chapter ⅩⅢ.Cycles on Powers of Quadrics
68.Split quadrics
69.Isomorphisms of quadrics
70.Isotropic quadrics
71.The Chow group of dimension O cycles on quadrics
72.The reduced Chow group
73.Cycles on X2
Chapter ⅪⅤ.The Izhboldin Dimension
74.The first Witt index of subforms
75.Correspondences
76.The main theorem
77.Addendum: The Pythagoras number
Chapter ⅩⅤ.Application of Steenrod Operations
78.Computation of Steenrod operations
79.Values of the first Witt index
80.Rost correspondences
81.On the 2-adic order of higher Witt indices, Ⅰ
82.Holes in In
83.On the 2-adic order of higher Witt indices, Ⅱ
84.Minimal height
Chapter ⅩⅥ.The Variety of Maximal Totally Isotropic Subspaces
85.The variety Gr(ψ)
86.The Chow ring of Gr(ψ)in the split case
87.The Chow ring of Gr(ψ)in the general case
……

本書是對二次型代數理論的全面研究，從古典理論到近期新進展，包括從未出版過的結果和證明。本書采用了代數幾何學的觀點，包括特征2的域上的二次型理論，證明盡可能是特征獨立的。對於一些結果，既給出了經典證明，又給出了幾何證明。本書第一部分包括經典的二次型和雙線性型代數理論，回答了該理論發展初期提出的許多問題。在代數幾何學隻有一門基礎課程的假設下，本書第二部分介紹了代數幾何學中必要的附加專題，包括Chow群理論、Chow運動和Steenrod運算。這些專題在第三部分中被用來發展二次型的現代幾何理論。