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W-Groups under Quadratic Extensions of Fields

  Published:2000-08-01
 Printed: Aug 2000
  • Ján Mináč
  • Tara L. Smith
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Abstract

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.
MSC Classifications: 11E81, 12D15 show english descriptions Algebraic theory of quadratic forms; Witt groups and rings [See also 19G12, 19G24]
Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) [See also 11Exx]
11E81 - Algebraic theory of quadratic forms; Witt groups and rings [See also 19G12, 19G24]
12D15 - Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) [See also 11Exx]
 

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