**CBMS Regional Conference Series in Mathematics**

Volume: 52;
1983;
143 pp;
Softcover

MSC: Primary 11;
Secondary 12

Print ISBN: 978-0-8218-0702-6

Product Code: CBMS/52

List Price: $36.00

Individual Price: $28.80

**Electronic ISBN: 978-1-4704-2414-5
Product Code: CBMS/52.E**

List Price: $36.00

Individual Price: $28.80

# Orderings, Valuations and Quadratic Forms

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*Tsit-Yuen Lam*

A co-publication of the AMS and CBMS

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 self-contained; 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 valuation-theoretic 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 all-important 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 valuation-theoretic Local-Global Principle for reduced quadratic forms.

#### Table of Contents

# Table of Contents

## Orderings, Valuations and Quadratic Forms

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface vii8 free
- Introduction ix10 free
- 1. The reduced theory of quadratic forms 114 free
- 2. Compatibility between valuations and orderings 1629
- 3. Compatibility between valuations and preorderings 2134
- 4. T-forms under a compatibile valuation 3245
- 5. Introduction to fans 3952
- 6. The Representaiton Problem: solution for fans 5063
- 7. The Representation Problem: reduction to fans 5770
- 8. The chain length of a preordering 6275
- 9. The real holomorphy ring and real-valued places 7386
- 10. Real-valued places associated to a preordering T 8093
- 11. The Prüfer ring A[sub(T)] associated with a preordering T 8598
- 12. The valuation ring A[sup(T)] associated with a preordering T 92105
- 13. Stability index of preorderings 99112
- 14. T-semiorderings 105118
- 15. Compatibility between valuations and T-semiorderings 111124
- 16. Pasch preorderings and their characterizations 119132
- 17. SAP preorderings and their characterizations 126139
- 18. An isotropy principle 133146
- References 138151
- Back Cover Back Cover1157