Special groups: Boolean-theoretic methods in the theory of quadratic forms: Boolean-Theoretic Methods in the Theory of Quadratic Forms
About this Title
M. A. Dickmann and F. Miraglia
Publication: Memoirs of the American Mathematical Society
Publication Year 2000: Volume 145, Number 689
ISBNs: 978-0-8218-2057-5 (print); 978-1-4704-0280-8 (online)
MathSciNet review: 1677935
MSC: Primary 11E81; Secondary 12D15, 19G12
This monograph presents a systematic study of Special Groups, a first-order universal-existential 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 Stiefel-Whitney invariants, the additive series—expressed in terms of meet and join—, which we call Horn-Tarski 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 Boolean-theoretic 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 model-theoretic study of the first-order theory of reduced special groups, where, amongst other things we determine its model-companion.
The first-order 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.
Table of Contents
- 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. Horn-Tarski and Stiefel-Whitney 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