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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