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# Une caractérisation des corps satisfaisant le théorème de l'axe principal

Published:1997-03-01
Printed: Mar 1997
• A. Movahhedi
• A. Salinier
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## Abstract

Resum\'e. On caract\'erise les corps $K$ satisfaisant le th\'eor\eme de l'axe principal \a l'aide de propri\'et\'es des formes carac\-t\'erisation de ces m\^emes corps due \a Waterhouse, on retrouve \a partir de l\`a, de fa\c{c}on \'el\'ementaire, un r\'esultat de Becker selon lequel un pro-$2$-groupe qui se r\'ealise comme groupe de Galois absolu d'un tel corps $K$ est engendr\'e par des involutions. ABSTRACT. We characterize general fields $K$, satisfying the Principal Axis Theorem, by means of properties of trace forms of the finite extensions of $K$. From this and Waterhouse's characterization of the same fields, we rediscover, in quite an elementary way, a result of Becker according to which a pro-$2$-group which occurs as the absolute Galois group of such a field $K$, is generated by
 MSC Classifications: 11E10 - Forms over real fields 12D15 - Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) [See also 11Exx]