The arithmetic theory of quadratic forms is a rich branch of number theory that has had important applications to several areas of pure mathematics—particularly group theory and topology—as well as to cryptography and coding theory. This book is a self-contained introduction to quadratic forms that is based on graduate courses the author has taught many times. It leads the reader from foundation material up to topics of current research interest—with special attention to the theory over the integers and over polynomial rings in one variable over a field—and requires only a basic background in linear and abstract algebra as a prerequisite. Whenever possible, concrete constructions are chosen over more abstract arguments. The book includes many exercises and explicit examples, and it is appropriate as a textbook for graduate courses or for independent study. To facilitate further study, a guide to the extensive literature on quadratic forms is provided.

* A brief classical introduction
* Quadratic spaces and lattices
* Valuations, local fields, and padic numbers
* Quadratic spaces over Qp
* Quadratic spaces over Q
* Lattices over principal ideal domains
* Initial integral results
* Local classification of lattices
* The local-global approach to lattices
* Lattices over Fq
* Applications to cryptography
* Further reading
* Bibliography
* Index