The Galois group of a Special Group

In this ongoing work, we extend to a class of well-behaved pre-special hyperfields the work of J. Min\'a c and Spira (minac1996witt) that describes a (pro-2) -group of a field extension that encodes the quadratic form theory of a given field F: in adem1999cohomology it is shown that its associated cohomology ring contains a copy of the cohomology ring of the field F. Our construction, a contravariant functor into the category of "pointed" pro-2-groups, is essentially given by generators and relations of profinite-2-groups. We prove that such profinite groups Gal (F) encode the space of orders of the special group canonically associated to the hyperfield F and provide a criterion to detect when F is formally real or not.