 
Summary: Realizing profinite reduced special groups
V. Astier, H. Mariano
1 Introduction
The theory of special groups is an axiomatization of the algebraic theory of
quadratic forms, introduced by Dickmann and Miraglia (see [4]). The class of
special groups, together with its morphisms, forms a category. As for other
such axiomatisations, the main examples of special groups are provided by
fields, in this case by applying the special group functor, which associates to
each field F a special group SG(F) describing the theory of quadratic forms
over F.
The category of special groups is equivalent to that of abstract Witt
rings via covariant functors, while the category of reduced special groups is
equivalent, via the restriction of the same covariant functors, to the category
of reduced abstract Witt rings (see [4, 1.25 and 1.26] ; recall that the special
group of a field F is reduced if and only if F is formally real and Pythagorean).
The category of reduced special groups is also equivalent, via contravariant
functors, to the category of abstract spaces of orderings (see [4, Chapter 3]).
The question whether it is possible to realize every (reduced) special
group as the special group of some (formally real, Pythagorean) field is still
open, but the case of finite reduced special groups (actually of reduced special
