**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.