To each field $F$ of characteristic not $2$, one can associate acertain Galois group $G_F$, the so-called W-group of $F$, whichcarries 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 quadraticextension of $F$. We obtain a precise description in the case when$F$ is a pythagorean formally real field and $a = -1$, and show thatthe W-group of a proper field extension $K/F$ is a subgroup of theW-group of $F$ if and only if $F$ is a formally real pythagorean fieldand $K = F(sqrt{-1})$. This theorem can be viewed as an analogue ofthe classical Artin-Schreier's theorem describing fields fixed byfinite subgroups of absolute Galois groups. We also obtain preciseresults in the case when $a$ is a double-rigid element in $F$. Someof these results carry over to the general setting.
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机译：对于每个特性不是$ 2 $的字段$ F $，可以将一定的Galois组$ G_F $关联，即所谓的$ F $的W-group，其携带的信息与Witt环$ W（F）$基本相同$ F $。在本文中，我们研究了$ wg $和$ G_ {F（sqrt {a}）} $之间的联系，其中$ F（sqrt {a}）$是$ F $的适当二次扩展。在$ F $是毕达哥拉斯形式上的实数场且$ a = -1 $的情况下，我们得到了一个精确的描述，并表明适当场扩展$ K / F $的W-群是W的W-群的子群。当且仅当$ F $是形式上真实的毕达哥拉斯域且$ K = F（sqrt {-1}）$时，$ F $。该定理可以看作是经典Artin-Schreier定理的类似物，该定理描述了由绝对Galois群的有限子群固定的场。当$ a $是$ F $中的双重刚性元素时，我们还可以获得精确的结果。这些结果中的一部分会延续到常规设置中。
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