Contents
IX
§6.10. Remarks on Lattices over More General Rings 141
Exercises 142
Chapter 7. Initial Integral Results 145
§7.1. The Minimum of a Lattice; Definite Binary Z-Lattices 146
§7.2. Hermite's Bound on min L, with a Supplement for &[x]-Lattices 149
§7.3. Djokovic's Reduction of A:[x]-Lattices; Harder's Theorem 153
§7.4. Finiteness of Class Numbers (The Anisotropic Case) 156
Exercises 158
Chapter 8. Local Classification of Lattices 161
§8.1. Jordan Splittings 161
§8.2. Nondyadic Classification 164
§8.3. Towards 2-adic Classification 165
Exercises 171
Chapter 9. The Local-Global Approach to Lattices 175
§9.1. Localization 176
§9.2. The Genus 178
§9.3. Maximal Lattices and the Cassels-Pfister Theorem 181
§9.4. Sums of Integer Squares (Second Version) 184
Exercises 187
§9.5. Indefinite Unimodular Z-Lattices 188
§9.6. The Eichler-Kneser Theorem; the Lattice Z
n
191
§9.7. Growth of Class Numbers with Rank 196
§9.8. Introduction to Neighbor Lattices 201
Exercises 205
Chapter 10. Lattices over ¥q[x] 207
§10.1. An Initial Example 209
§10.2. Classification of Definite ¥q[^-Lattices 210
§10.3. On the Hasse-Minkowski Theorem over ¥q(x) 218
§10.4. Representation by F9[x]-Lattices 220
Exercises 223
Chapter 11. Applications to Cryptography 225
§11.1. A Brief Sketch of the Cryptographic Setting 225
811.2. Lattices in
Rn
227
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