W-Groups under Quadratic Extensions of Fields

Mináč, Ján ; Smith, Tara L.

Canadian journal of mathematics, Tome 52 (2000) no. 4, pp. 833-848

Résumé

To each field $F$ of characteristic not 2, one can associate a certain Galois group ${{\mathcal{G}}_{F}}$ , the so-called $\text{W}$ -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 ${{\mathcal{G}}_{F}}$ and ${{\mathcal{G}}_{F(\sqrt{a})}}$ , 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 $\text{W}$ -group of a proper field extension $K/F$ is a subgroup of the $\text{W}$ -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.

Détail

DOI : 10.4153/CJM-2000-036-0

Mots-clés : 11E81, 12D15

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