This book grew out of a graduate course that I gave at the University of Cambridge in the Easter Term of 1993. The idea of publishing a somewhat enlarged and polished version of my lectures came from Professor J.W.S. Cassels, who, in addition, made it possible for me to spend my sabbatical at Cambridge supported by a Research Grant of the SERC and an appointment as a “Visiting Fellow Commoner” of Trinity College. I thank these institutions for their help in making my stay in Cambridge a very pleasant one.

I should point out in this connection that a great deal of my research on quadratic forms began in the year 1963 when I attented a colloquium talk given by Cassels on “Sums of Squares of Rational Functions” at the University of Göttingen. Later, our connections intensified during the Academic Year 1966/67 when I studied and lectured in Cambridge. My early Lecture Notes [Pfister 19671] give an idea of the status of the algebraic theory of quadratic forms in those days. Thus, much of my previous work as well as the present book owe their existence to the constant encouragement and interest of Cassels over many years. For this reason, I wish to express my deep gratitude to him.

This book is not a systematic treatise on quadratic forms. Excellent books of this kind are already available, in particular the books of O'Meara [O'M] on the arithmetic theory over number fields and their integer domains and the books of Lam [L] and Scharlau [S] on the algebraic theory over general fields.