Witt rings and the quaternion group (Q1265382)

Let \(F\) be a formally real field having at most a finite number of orderings. This article examines the structure of the Witt ring of \(F\) and the maximal pro-2 Galois group of \(F\) in the case when \(F\) admits no quaternionic Galois extension. This extends earlier work of the author where the case of a field having at most one ordering and no quaternionic Galois extensions was considered. Finally, the author obtains an appropriate analogue for the case of an odd prime \(p\), where \(F\) contains a primitive \(p\)th root of unity and \(| F/F^p| > p\).