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. Miná\v c and Spira (\cite{minac1996witt}) that describes a (pro-2)-group of a field extension that encodes the quadratic form theory of a given field $F$: in \cite{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.