Formally real fields from a Galois theoretic perspective (Q1963987)

Let \(F\) be a field of characteristic not 2. Let \(F^{(3)}\) be the compositum over \(F\) of all cyclic of order 2, cyclic of order 4 and dihedral of order 8 extensions of \(F\). Then the Galois group of \(F^{(3)}\) over \(F\) is called the \(W\)-group \({\mathcal{G}}_F\) of \(F\). The authors investigate how the lattice of orderings and of preorderings on \(F\) is determined by \({\mathcal{G}}_F\). There is a close connection between orderings and nontrivial involutions in \({\mathcal{G}}_F\). The Galois-theoretic manifestation of the preordering on \(F\) is an essential subgroup of \({\mathcal{G}}_F\) generated by involutions. By an essential subgroup, the authors mean a closed subgroup \(H\) such that the Frattini subgroup of \(H\) is given by the intersection of \(H\) with the Frattini subgroup of \(G\).