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