Formally real fields from a Galois theoretic perspective Original Research Article

To each field F of characteristic not 2, one can associate a certain Galois group image, the so-called W-group of F, which carries essentially the same information as the Witt ring W(F) of equivalence classes of anisotropic quadratic forms over F. There is a close connection between (nontrivial) involutions in image and orderings on F. The purpose of this paper is to investigate how the lattice of orderings and preorderings on F is determined by image, and to provide a Galois-theoretic version of reduced Witt rings.