目錄
Prerequisites and Notation
Part One Arithmetic Theory of Fields
Chapter I. Valuated Fields
11. Valuations
12. Arehimedean valuations
13. Non-archimedean valuations
14. Prolongation of a complete valuation to a finite extension
15. Prolongat/on of any valuation to a finite separable extension . .
16. Discrete valuations
Chapter II. Dedekind Theory of Ideals
21. Dedekind axioms for S
22. Ideal theory
23. Extens/on fields
Chapter III. Fields of Number Theory
31. Rational global fields
32. Local fields
33. Global fields
Part Two Abstract Theory of Quadratic Forms
Chapter IV. Quadratic Forms and the O~ogonal Group
41. Forms, matrices and spaces
42. Quadratic spaces
43. Special subgroups of O.(V)
Chapter V. The Algebras of Quadratic Forms
Sl. Tensor products
52. Wedderburn's theorem on central simple algebras
53. Extending the field of scalars
54, The Clifford algebra
55. The spinor norm
56. Special subgroups of O,(V)
57. Quaternion algebras
58. The Hasse algebra
Part Three Arithmetic Theory of Quadratic Forms over Fields
Chapter VI. The Equivalence of Quadratic Forms
61. Complete archimedean fields
62. Finite fields
63. Local fields
64. Global notation
68. Squares and norms in giobal fields
66. Quadratic forms over global fields
Chapter VII. Hilbert's Reciprocity Law
71. Proof of the reciprocity law
72. Existence of forms with prescribed local behavior
73. The quadratic reciprocity law
Part Four Arithmetic Theory of ~uadratie Forms over Rings
Chapter VIII. Quadratic Forms over Dedekind Domains
81. Abstract lattices
82. Lattices in quadratic spaces
Chapter IX. Integral Theory of Quadratic Forms over Local Fields
91. Generalities
92. Classification of lattices over non-dyadic fields
93. Classification of lattices over dyadic fields
94. Effective determination of the invariants
95. Special subgroups of 0. (V)
Chapter X. Integral Theory of Quadratic Forms over Global Fields
101. Elementary properties of the orthogonal group over arithmetic fields
102. The genus and the spinor genus
103. Finiteness of class number
104. The class and the spinor genus in the indefinite case
10S. The indecomposable splitting of a definite lattice
106. Definite unimodular lattices over the rational integers
Bibliography
Index
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