Memoirs of the American Mathematical Society 2000; 247 pp; softcover Volume: 145 ISBN10: 0821820575 ISBN13: 9780821820575 List Price: US$72 Individual Members: US$43.20 Institutional Members: US$57.60 Order Code: MEMO/145/689
 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 seriesexpressed in terms of meet and symmetric differenceconstitutes a Boolean version of the StiefelWhitney invariants, the additive seriesexpressed 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. Readership Graduate students and research mathematicians interested in number theory. Table of Contents  Special groups
 Pfister forms, saturated subgroups and quotients
 The space of orders of a reduced group. Duality
 Boolean algebras and reduced special groups
 Embeddings
 Special groups of continuous functions
 HornTarski and StiefelWhitney invariants
 Algebraic Ktheory of fields and special groups
 Marshall's conjecture for pythagorean fields
 The category of special groups
 Some model theory of special groups
 Appendix A. The universal theory of reduced special groups
 Appendix B. Table of references for [DM1] and [DM2]
 Bibliography
 Index
