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
Previous Page Next Page