To each field $F$ of characteristic not $2$, one can associate a certain Galois group $\G_F$, the so-called W-group 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 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 $K = F(\sqrt{-1})$. 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.