Abstract view
WGroups under Quadratic Extensions of Fields


Published:20000801
Printed: Aug 2000
Abstract
To each field $F$ of characteristic not $2$, one can associate a
certain Galois group $\G_F$, the socalled Wgroup of $F$, which
carries essentially the same information as the Witt ring $W(F)$ of
$F$. In this paper we investigate the connection between $\wg$ and
$\G_{F(\sqrt{a})}$, where $F(\sqrt{a})$ 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 Wgroup of a proper field extension $K/F$ is a subgroup of the
Wgroup of $F$ if and only if $F$ is a formally real pythagorean field
and $K = F(\sqrt{1})$. This theorem can be viewed as an analogue of
the classical ArtinSchreier's theorem describing fields fixed by
finite subgroups of absolute Galois groups. We also obtain precise
results in the case when $a$ is a doublerigid element in $F$. Some
of these results carry over to the general setting.