Formally real fields, pythagorean fields, C-fields and W-groups (Q583279)

In previous work the authors showed that under some mild condition on the field F the Galois group \({\mathcal G}_ F\) of a certain 2-extension \(F^{(3)}\) of F completely determines the Witt ring of F. In this paper it is shown that the presence or absence of certain involutions in \({\mathcal G}_ F\) determines whether F is a formally real field or not. More precisely a 1-1 correspondence is shown between orderings of fields and certain classes of involutions in \({\mathcal G}_ F\). The relative real closure of a formally real field F inside of \(F^{(3)}\) is defined and this is used to give a new characterization of superpythagorean fields. Moreover a characterization of pythagorean fields via \({\mathcal G}_ F\) is shown as well as the description of all possible Abelian groups \({\mathcal G}_ F\). The main tools are dihedral and \({\mathbb{Z}}/4 {\mathbb{Z}}\) extensions, together with simple Galois theory. This paper can be viewed as transfer of classical results of Artin- Schreier and Becker to much smaller fields.