Realizing profinite reduced special groups

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 groups of finite chain length) has been positively answered by the combination of two results of Kula and Marshall: In [10], Kula showed that the product of two finite special groups of (formally real, Pythagorean) fields is still the special group of some (formally real, Pythagorean) field, and in [14], Marshall showed that every finite reduced special group can be constructed from the special group of any real closed field by applying a finite number of times the operations of product and extension (Marshall’s result is actually stated and proved for abstract spaces of orderings). Since the extension of the special group of a (formally real, Pythagorean) field is still the special group of a (formally real, Pythagorean) field, it shows that every finite reduced special group (or reduced special group of finite chain length) is realized as the special group of a field.