eBook ISBN:  9781470424145 
Product Code:  CBMS/52.E 
List Price:  $38.00 
Individual Price:  $30.40 
eBook ISBN:  9781470424145 
Product Code:  CBMS/52.E 
List Price:  $38.00 
Individual Price:  $30.40 

Book DetailsCBMS Regional Conference Series in MathematicsVolume: 52; 1983; 143 ppMSC: Primary 11; Secondary 12
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

Chapters

1. The reduced theory of quadratic forms

2. Compatibility between valuations and orderings

3. Compatibility between valuations and preorderings

4. $T$forms under a compatibile valuation

5. Introduction to fans

6. The Representation Problem: solution for fans

7. The Representation Problem: reduction to fans

8. The chain length of a preordering

9. The real holomorphy ring and realvalued places

10. Realvalued places associated to a preordering $T$

11. The Prüfer ring $A_T$ associated with a preordering $T$

12. The valuation ring $A^T$ associated with a preordering $T$

13. Stability index of preorderings

14. $T$semiorderings

15. Compatibility between valuations and $T$semiorderings

16. Pasch preorderings and their characterizations

17. SAP preorderings and their characterizations

18. An isotropy principle


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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.

Chapters

1. The reduced theory of quadratic forms

2. Compatibility between valuations and orderings

3. Compatibility between valuations and preorderings

4. $T$forms under a compatibile valuation

5. Introduction to fans

6. The Representation Problem: solution for fans

7. The Representation Problem: reduction to fans

8. The chain length of a preordering

9. The real holomorphy ring and realvalued places

10. Realvalued places associated to a preordering $T$

11. The Prüfer ring $A_T$ associated with a preordering $T$

12. The valuation ring $A^T$ associated with a preordering $T$

13. Stability index of preorderings

14. $T$semiorderings

15. Compatibility between valuations and $T$semiorderings

16. Pasch preorderings and their characterizations

17. SAP preorderings and their characterizations

18. An isotropy principle