§15§This book is intended to give an introduction to some parts of the algebraic theory of quadratic forms. It grew out of a course, given by the first author at the University of Saarbrucken during the academic year 1971/72. This course considered "weak methods" in the theory of quadratic forms, i.e. methods, which apply to the classes of forms in the Wittring rather than to the forms themselves. In the meantime some topics of this course (cf.VIII ) have been treated in an excellent way in other books - we only mention Lam G D . The part of the course covered by the present book is the ringtheoretic approach to the structure of Wittrings in the framework of "abstract" Wittrings, based on the papers [)f[], [)!J. Working with these rings yields a greater flexibility than just working with the Wittrings of fields. Besides this the study of abstract Wittrings has the advantage, that the results are not limited to the case of Wittrings over fields, but carryover to local rings, and thus g §15§ive an indespensable contribution to the local part of a theory of forms over algebraic varieties (cf. [)~ for a first impression of this theory). After having read this book, the reader will be well prepared to understand the local theory (cf. e. g. Q![]). §03§I: Basic facts about symmetric bilinear forms, and definition of the Wittring..-§1 Bilinear Spaces.-§2 Witt- and Grothendieck-rings.- Appendix: Quadratic Forms.- II: The structure of Wittrings.-§1 Generators and Relations.-§2 The prime ideals of a Wittring.-§3 Nilpotent and torsion elements.-§4 Application: The theorem of Artin-Pfister.-§5 Complements to the structure theory.-§6 Characterization of abstract Wittrings.-§7 Fields with isomorphic Wittrings.- III: Reduced Wittrings.-§1 Von Neumann regular rings.-§2 Topological description of reduced Wittrings.-§3 A Nullstellensatz for Witt ideals and a generalization of the theorem of Artin-Pfister.-§4 When are Wittrings group rings?.-§5 Fields with strong approximation for orderings.- References.
