Basic Quadratic Forms

by Larry J. Gerstein

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.

Genres: Mathematics

255 pages, Hardcover

First published April 2, 2008

Reviews

[Spunkyblah · Rating 5 out of 5]

I'm a bit biased since I helped Prof. Gerstein with some proof-reading and light editing in the early stages of the book (no, I don't get anything out of sales of this book), but I think the text turned out wonderfully. The subject of quadratic forms is covered from the very beginnings (with definitions of pretty much anything that one would not already know from linear algebra and a introductory abstract algebra series) through fun/interesting diversionslike representations of prime numbers up to topics concerning such specific and recent topics as diagonalization of lattices. This book is readable (much better, in my opinion, than O'Meara's text, which is considered a bit of a canon on the topic and was the book I had to struggle through when learning the topics that are not covered in this text) and has many exercises that are aimed at helping the reader learn the subject. It is surprising how many graduate texts seem to be written without the goal of helping students learn. In any case, if you are interested in a subject that mixes linear and abstract algebra with number theory, it may be useful to take a look at this book. If you are interested in cryptography based on lattice theory, I recommend checking out the later chapters of this textbook. This book is aimed at those who are just starting to learn about the details of quadratic forms, so if you find yourself referring to other QF books as scary, dense, or unnecessarily cryptic, you may want to check this book out.