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Contents

Preface

Chapter 1. A Brief Classical Introduction

§1.1. Quadratic Forms as Polynomials

§1.2. Representation and Equivalence; Matrix Connections; Discriminants

Exercises

§1.3. A Brief Historical Sketch, and Some References to the Literature

Chapter 2. Quadratic Spaces and Lattices

§2.1. Fundamental Definitions

§2.2. Orthogonal Splitting; Examples of Isometry and Non-isometry

Exercises

§2.3. Representation, Splitting, and Isotropy; Invariants u(F) and s(F)

§2.4. The Orthogonal Group of a Space

§2.5. Witt's Cancellation Theorem and Its Consequences

§2.6. Witt's Chain Equivalence Theorem

§2.7. Tensor Products of Quadratic Spaces; the Witt ring of a field

Exercises

§2.8. Quadratic Spaces over Finite Fields

§2.9. Hermitian Spaces

Exercises

Chapter 3. Valuations, Local Fields, and p-adic Numbers

§3.1. Introduction to Valuations

Exercises

§3.2. Equivalence of Valuations; Prime Spots on a Field

§3.3. Completions, Q[sub(p)], Residue Class Fields

§3.4. Discrete Valuations

§3.5. The Canonical Power Series Representation

§3.6. Hensel's Lemma, the Local Square Theorem, and Local Fields

§3.7. The Legendre Symbol; Recognizing Squares in Q[sub(p)]

Exercises

Chapter 4. Quadratic Spaces over Q[sub(p)]

§4.1. The Hilbert Symbol

§4.2. The Hasse Symbol (and an Alternative)

§4.3. Classification of Quadratic Q[sub(p)]-Spaces

§4.4. Hermitian Spaces over Quadratic Extensions of Q[sub(p)]

Exercises

Chapter 5. Quadratic Spaces over Q

§5.1. The Product Formula and Hilbert's Reciprocity Law

§5.2. Extension of the Scalar Field

§5.3. Local to Global: The Hasse-Minkowski Theorem

§5.4. The Bruck-Ryser Theorem on Finite Projective Planes

§5.5. Sums of Integer Squares (First Version)

Exercises

Chapter 6. Lattices over Principal Ideal Domains

§6.1. Lattice Basics

§6.2. Valuations and Fractional Ideals

§6.3. Invariant factors

§6.4. Lattices on Quadratic Spaces

§6.5. Orthogonal Splitting and Triple Diagonalization

§6.6. The Dual of a Lattice

Exercises

§6.7. Modular Lattices

§6.8. Maximal Lattices

§6.9. Unimodular Lattices and Pythagorean Triples

§6.10. Remarks on Lattices over More General Rings

Exercises

Chapter 7. Initial Integral Results

§7.1. The Minimum of a Lattice; Definite Binary Z-Lattices

§7.2. Hermite's Bound on min L, with a Supplement for k[x]-Lattices

§7.3. Djokovic's Reduction of k[x]-Lattices; Harder's Theorem

§7.4. Finiteness of Class Numbers (The Anisotropic Case)

Exercises

Chapter 8. Local Classification of Lattices

§8.1. Jordan Splittings

§8.2. Nondyadic Classification

§8.3. Towards 2-adic Classification

Exercises

Chapter 9. The Local-Global Approach to Lattices

§9.1. Localization

§9.2. The Genus

§9.3. Maximal Lattices and the Cassels–Pfister Theorem

§9.4. Sums of Integer Squares (Second Version)

Exercises

§9.5. Indefinite Unimodular Z-Lattices

§9.6. The Eichler-Kneser Theorem; the Lattice Z[sup(n)]

§9.7. Growth of Class Numbers with Rank

§9.8. Introduction to Neighbor Lattices

Exercises

Chapter 10. Lattices over F[sub(q)][x]

§10.1. An Initial Example

§10.2. Classification of Definite F[sub(q)][x]-Lattices

§10.3. On the Hasse-Minkowski Theorem over F[sub(q)][x]

§10.4. Representation by F[sub(q)][x]-Lattices

Exercises

Chapter 11. Applications to Cryptography

§11.1. A Brief Sketch of the Cryptographic Setting

811.2. Lattices in R[sup(n)]

§11.3. LLL-Reduction

§11.4. Lattice Attacks on Knapsack Cryptosystems

§11.5. Remarks on Lattice-Based Cryptosystems

Appendix: Further Reading

Bibliography

Index

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

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