CBMS Regional Conference Series in Mathematics 1983; 143 pp; softcover Number: 52 Reprint/Revision History: reprinted with corrections 1996 ISBN10: 0821807021 ISBN13: 9780821807026 List Price: US$36 Member Price: US$28.80 All Individuals: US$28.80 Order Code: CBMS/52
 The remarkable relationships and interplay between orderings, valuations and quadratic forms have been the object of intensive and fruitful study in recent mathematical literature. In this book, the author, a Steele Prize winner in 1982, provides an authoritative and beautifully written account of recent developments in the theory of the "reduced" Witt ring of a formally real field. This area of mathematics is growing rapidly and promises to become of increasing importance in reality questions in algebraic geometry. The book covers many results from original research papers published in the last fifteen years. The presentation in these notes is largely selfcontained; the only prerequisite might be a good working knowledge of general valuation theory and some familiarity with the basic notions and terminology of quadratic form theory. The first chapters of the author's previous book, published by W. A. Benjamin, are a good source for such background material. However, this volume may be read as an independent introduction to ordered fields and reduced quadratic forms using valuationtheoretic techniques. Orderings and valuations are related through the notion of compatibility; valuations and quadratic forms are related through the notion of residue forms, while quadratic forms and orderings are related through the notion of signatures. After a beginning chapter on the reduced theory of quadratic forms, the author lays the foundation for the study of compatibility. This is followed by an introduction to the techniques of residue forms and the relevant Springer theory. The author then presents the solution of the Representation Problem due to Bechker and Bröcker, with simplifications due to Marshall. The notion of fans plays an allimportant role in this approach. Further chapters threat the theory of real places and the real holomorphy ring, prove Bröcker's theorem on the trivialization of fans, and study in detail two important invariants of a preordering (the chain length and the stability index). Other topics treated include the notion of semiorderings, its applications to SAP fields and SAP preorderings, and the valuationtheoretic LocalGlobal Principle for reduced quadratic forms. Readership Table of Contents  The reduced theory of quadratic forms
 Compatibility between valuations and orderings
 Compatibility between valuations and preorderings
 Appendix: Henselian fields and 2Henselian fields
 \(T\)forms under a compatible valuation
 Introduction to fans
 Appendix: Superpythagorean fields
 The representation problem: solution for fans
 The representation problem: reduction to fans
 The chain length of a preordering
 The real holomorphy ring and realvalued places
 Realvalued places associated to a preordering \(t\)
 The Prüfer ring \(A_T\) associated with a preordering \(t\)
 The valuation ring \(A^T\) associated with a preordering \(t\)
 Stability index of preorderings
 \(T\)semiorderings
 Compatibility between valuations and \(T\)semiorderings
 Pasch preorderings and their characterizations
 SAP preorderings and their characterizations
 An isotropy principle
 References
