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Contents
Chapter 3. Valuations, Local Fields, and p-adic Numbers 51
§3.1. Introduction to Valuations 51
§3.2. Equivalence of Valuations; Prime Spots on a Field 54
Exercises 58
§3.3. Completions, Qp, Residue Class Fields 59
§3.4. Discrete Valuations 63
§3.5. The Canonical Power Series Representation 64
§3.6. Hensel's Lemma, the Local Square Theorem, and Local Fields 69
§3.7. The Legendre Symbol; Recognizing Squares in Qp 74
Exercises 76
Chapter 4. Quadratic Spaces over Qp 81
§4.1. The Hilbert Symbol 81
§4.2. The Hasse Symbol (and an Alternative) 86
§4.3. Classification of Quadratic Qp-Spaces 87
§4.4. Hermitian Spaces over Quadratic Extensions of Qp 92
Exercises 94
Chapter 5. Quadratic Spaces over Q 97
§5.1. The Product Formula and Hilbert's Reciprocity Law 97
§5.2. Extension of the Scalar Field 98
§5.3. Local to Global: The Hasse-Minkowski Theorem 99
§5.4. The Bruck-Ryser Theorem on Finite Projective Planes 105
§5.5. Sums of Integer Squares (First Version) 109
Exercises 111
Chapter 6. Lattices over Principal Ideal Domains 113
§6.1. Lattice Basics 114
§6.2. Valuations and Fractional Ideals 116
§6.3. Invariant factors 118
§6.4. Lattices on Quadratic Spaces 122
§6.5. Orthogonal Splitting and Triple Diagonalization 124
§6.6. The Dual of a Lattice 128
Exercises 130
§6.7. Modular Lattices 133
§6.8. Maximal Lattices 136
§6.9. Unimodular Lattices and Pythagorean Triples 138
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