Xl l

Preface

the reader will be exposed to topics of current research interest. I will be

happy if the readers find themselves wanting to pursue some aspects of the

subject in more detail than this book can provide; accordingly, I will offer

some references to the literature and recommendations for further study.

Before 1937, quadratic forms were treated primarily as homogeneous

polynomials of degree 2 acted on by transformations that could change a

given quadratic form into certain other ones. (And a fundamental question

was: into which other ones?) But a pioneering paper by Witt in 1937 brought

a more geometric flavor to the subject, putting it on the border of linear

algebra and number theory—roughly speaking, a theory of generalized inner

products on modules. Our coefficient ring of interest will most often be the

ring Z of rational integers, though we will also give special attention to the

polynomial rings Fg[x]. (Here ¥q denotes a finite field with q elements.) We

will see that before we can effectively explore quadratic forms over a given

domain R, we may need to extend i?, perhaps in many ways, to larger rings.

The extended domains (specifically, the p-adic number fields, their rings of

integers, and their function-field analogues) may possess complications of

their own that require clarification before we can consider quadratic forms

over them; but once we have achieved that clarification, we may find that

quadratic forms over those extensions are far more tractable than over R.

When that happens, the trick is to then bring that information down to R

and apply it to the original forms.

This book has evolved from lecture notes for introductory graduate

courses on quadratic forms I have taught many times at the University of

California, Santa Barbara, and once at Dartmouth College. Typically these

courses have been populated by second-year graduate students who have

already had a basic course in algebraic structures, and this is the primary

audience I have had in mind during the writing process. But in fact the book

should be readable by anyone with a strong undergraduate background in

linear and abstract algebra who has also seen the construction of the real

numbers from the rationals.

Naturally the contents of this book have been shaped by my own inter-

ests, experience, and tastes, and I have no doubt that some mathematicians

will lament the absence of one or more of their favorite topics in the theory

of quadratic forms. But I hope that their concerns will be eased by seeing in

these pages some new perspectives—and occasionally something completely

new—and that where the material is familiar they will experience the joy of

revisiting old friends.

I thank Miklos Ajtai, Mark Gaulter, Arnold Johnson, Timothy O'Meara,

Martin Scharlemann, Thomas Shemanske, and the anonymous referees for

their helpful comments, and I especially thank Melissa Flora for her detailed