eBook ISBN:  9781470402808 
Product Code:  MEMO/145/689.E 
List Price:  $72.00 
MAA Member Price:  $64.80 
AMS Member Price:  $43.20 
eBook ISBN:  9781470402808 
Product Code:  MEMO/145/689.E 
List Price:  $72.00 
MAA Member Price:  $64.80 
AMS Member Price:  $43.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 145; 2000; 247 ppMSC: Primary 11; 06; Secondary 12; 03
This monograph presents a systematic study of Special Groups, a firstorder universalexistential axiomatization of the theory of quadratic forms, which comprises the usual theory over fields of characteristic different from 2, and is dual to the theory of abstract order spaces.
The heart of our theory begins in Chapter 4 with the result that Boolean algebras have a natural structure of reduced special group. More deeply, every such group is canonically and functorially embedded in a certain Boolean algebra, its Boolean hull. This hull contains a wealth of information about the structure of the given special group, and much of the later work consists in unveiling it.
Thus, in Chapter 7 we introduce two series of invariants “living” in the Boolean hull, which characterize the isometry of forms in any reduced special group. While the multiplicative series—expressed in terms of meet and symmetric difference—constitutes a Boolean version of the StiefelWhitney invariants, the additive series—expressed in terms of meet and join—, which we call HornTarski invariants, does not have a known analog in the field case; however, the latter have a considerably more regular behaviour. We give explicit formulas connecting both series, and compute explicitly the invariants for Pfister forms and their linear combinations.
In Chapter 9 we combine Booleantheoretic methods with techniques from Galois cohomology and a result of Voevodsky to obtain an affirmative solution to a long standing conjecture of Marshall concerning quadratic forms over formally real Pythagorean fields.
Boolean methods are put to work in Chapter 10 to obtain information about categories of special groups, reduced or not. And again in Chapter 11 to initiate the modeltheoretic study of the firstorder theory of reduced special groups, where, amongst other things we determine its modelcompanion.
The firstorder approach is also present in the study of some outstanding classes of morphisms carried out in Chapter 5, e.g., the pure embeddings of special groups. Chapter 6 is devoted to the study of special groups of continuous functions.
ReadershipGraduate students and research mathematicians interested in number theory.

Table of Contents

Chapters

1. Special groups

2. Pfister forms, saturated subgroups and quotients

3. The space of orders of a reduced group. Duality

4. Boolean algebras and reduced special groups

5. Embeddings

6. Special groups of continuous functions

7. HornTarski and StiefelWhitney invariants

8. Algebraic $K$theory of fields and special groups

9. Marshall’s conjecture for Pythagorean fields

10. The category of special groups

11. Some model theory of special groups


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This monograph presents a systematic study of Special Groups, a firstorder universalexistential axiomatization of the theory of quadratic forms, which comprises the usual theory over fields of characteristic different from 2, and is dual to the theory of abstract order spaces.
The heart of our theory begins in Chapter 4 with the result that Boolean algebras have a natural structure of reduced special group. More deeply, every such group is canonically and functorially embedded in a certain Boolean algebra, its Boolean hull. This hull contains a wealth of information about the structure of the given special group, and much of the later work consists in unveiling it.
Thus, in Chapter 7 we introduce two series of invariants “living” in the Boolean hull, which characterize the isometry of forms in any reduced special group. While the multiplicative series—expressed in terms of meet and symmetric difference—constitutes a Boolean version of the StiefelWhitney invariants, the additive series—expressed in terms of meet and join—, which we call HornTarski invariants, does not have a known analog in the field case; however, the latter have a considerably more regular behaviour. We give explicit formulas connecting both series, and compute explicitly the invariants for Pfister forms and their linear combinations.
In Chapter 9 we combine Booleantheoretic methods with techniques from Galois cohomology and a result of Voevodsky to obtain an affirmative solution to a long standing conjecture of Marshall concerning quadratic forms over formally real Pythagorean fields.
Boolean methods are put to work in Chapter 10 to obtain information about categories of special groups, reduced or not. And again in Chapter 11 to initiate the modeltheoretic study of the firstorder theory of reduced special groups, where, amongst other things we determine its modelcompanion.
The firstorder approach is also present in the study of some outstanding classes of morphisms carried out in Chapter 5, e.g., the pure embeddings of special groups. Chapter 6 is devoted to the study of special groups of continuous functions.
Graduate students and research mathematicians interested in number theory.

Chapters

1. Special groups

2. Pfister forms, saturated subgroups and quotients

3. The space of orders of a reduced group. Duality

4. Boolean algebras and reduced special groups

5. Embeddings

6. Special groups of continuous functions

7. HornTarski and StiefelWhitney invariants

8. Algebraic $K$theory of fields and special groups

9. Marshall’s conjecture for Pythagorean fields

10. The category of special groups

11. Some model theory of special groups