W-Groups under Quadratic Extensions of Fields

Abstract

AbstractTo each field F of characteristic not 2, one can associate a certain Galois group , the so-called W-group of F, which carries essentially the same information as the Witt ring W(F) of F. In this paperwe investigate the connection between and (√a), where F(√a) is a proper quadratic extension of F. We obtain a precise description in the case when F is a pythagorean formally real field and a = −1, and show that the W-group of a proper field extension K/F is a subgroup of the W-group of F if and only if F is a formally real pythagorean field and K = F(√−1). This theorem can be viewed as an analogue of the classical Artin-Schreier’s theorem describing fields fixed by finite subgroups of absolute Galois groups. We also obtain precise results in the case when a is a double-rigid element in F. Some of these results carry over to the general setting.