CONTENTS Introduction.................................................................................................................................. 5 Chapter I. Preliminaries oil round and Pfister forms.......................................................... 8 Chapter II. Basic properties of the Grothendieck ring......................................................... 11 § 1. Prime ideals in G(F)...................................................................................................... 11 § 2. Elements of special types............................................................................................ 15 § 3. Local properties of G(F)................................................................................................ 20 Chapter III. Group structure of G(F) and W(F)....................................................................... 23 §1. General decomposition theorems.............................................................................. 23 § 2. Value sets of binary forms and the group structure of G(F) and W(F)................. 28 § 3. The rank of G(F) and W(F)............................................................................................ 32 Chapter IV. Equivalence of fields with respect to quadratic forms.................................... 36 § 1. G-equivalences.............................................................................................................. 36 § 2. W-equivalences.............................................................................................................. 41 § 3. Comparisons.................................................................................................................. 43 Chapter V. A Galois correspondence in the quadratic form theory.................................. 46 § 1. The binary case............................................................................................................. 46 § 2. A generalization.............................................................................................................. 54 Chapter VI. Field constructions................................................................................................ 57 Chapter VII. Open problems..................................................................................................... 60 References.................................................................................................................................. 62