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Some model-theoretic results in the algebraic theory of quadratic forms

Summary: Some model-theoretic results in the algebraic
theory of quadratic forms
Vincent Astier
1 Introduction
The theory of Witt rings of fields has led to different axiomatizations of quadratic
form theory, for example abstract Witt rings, quadratic form schemes, quater-
nionic structures; each of these axiomatizations highlighting a particular point
of view.
Recently, Dickmann and Miraglia have introduced a new axiomatization,
the theory of special groups, which they have developed to get new results: see
[DM], and [DM2] in which Marshall's and Lam's conjectures are proven.
The category of special groups (with its morphisms) is naturally isomorphic to
that of abstract Witt rings. Moreover, the theory of special groups is axioma-
tized by a finite set of formulae in a first-order language, and it is thus natural
to look at it form the point of view of model theory.
This is the subject matter of this paper, in which we present some model-
theoretic results concerning special groups of finite type (see definition 4.3):
We recall Feferman and Vaught's notion of generalized product and prove some
results about them, and we introduce briefly the main concepts concerning spe-
cial groups (sections 2 and 3).


Source: Astier, Vincent - Department of Mathematics, University College Dublin


Collections: Mathematics