Introduction to quadratic forms/ Timothy O.O'Meara

Chapter I. Valuated Fields 1
11. Valuations 1
12. Archimedean valuations 14
13. Non-archimedean valuations 20
14. Prolongation of a complete valuation to a finite extension 28
15.
Prolongation of any valuation to a finite separable extension .... 30
16. Discrete valuations 37
Chapter II. Dedekind Theory of Ideals 41
21. Dedekind axioms for S 42
22. Ideal theory 44
23. Extension fields 52
Chapter III. Fields of Number Theory 54
31. Rational global fields 54
32. Local fields 59
33. Global fields 65
Part Two
Abstract Theory of Quadratic Forms
Chapter IV. Quadratic Forms and the Orthogonal Group 82
41. Forms, matrices and spaces 82
42. Quadratic spaces 88
43. Special subgroups of 0„(F) 100
Chapter V. The Algebras of Quadratic Forms 112
51. Tensor products 113
52. Wedderburn's theorem on central simple algebras 118
53. Extending the field of scalars 129
54. The Clifford algebra 131
55. The spinor norm 137
56. Special subgroups of 0„(F) 141
57. Quaternion algebras 142
58. The Hasse algebra 149
XII Contents
Part Three
Arithmetic Theory of Quadratic Forms over Fields
Chapter VI. The Equivalence of Quadratic Forms 154
61. Complete archimedean fields 154
62. Finite fields 157
63. Local fields 158
64. Global notation 172
65. Squares and norms in global fields 173
66. Quadratic forms over global fields 186
Chapter VII. Hilbert's Reciprocity Law 190
71. Proof of the reciprocity law 190
72. Existence of forms with prescribed local behavior 203
73. The quadratic reciprocity law 205
Part Four
Arithmetic Theory of Quadratic Forms over Rings
Chapter VIII. Quadratic Forms over Dedekind Domains 208
81. Abstract lattices 208
82. Lattices in quadratic spaces 220
Chapter IX. Integral Theory of Quadratic Forms over Local Fields 239
91. Generalities 239
92. Classification of lattices over non-dyadic fields 246
93. Classification of lattices over dyadic fields 250
94. Effective determination of the invariants 279
95. Special subgroups of 280
Chapter X. Integral Theory of Quadratic Forms over Global Fields 284
101. Elementary properties of the orthogonal group over arithmetic fields 285
102. The genus and the spinor genus 297
103. Finiteness of class number 305
104. The class and the spinor genus in the indefinite case 311
105. The indecomposable splitting of a definite lattice 321
106. Definite unimodular lattices over the rational integers 323