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

Published online by Cambridge University Press:  20 November 2018

A. Movahhedi  and
A. Salinier
A. Movahhedi
Affiliation:
LACO (URA 1586 CNRS) Département de Mathématiques Faculté des Sciences 123, avenue Albert Thomas 87060 Limoges Cedex France
A. Salinier
Affiliation:
LACO (URA 1586 CNRS) Département de Mathématiques Faculté des Sciences 123, avenue Albert Thomas 87060 Limoges Cedex France
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Résumé

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On caractérise les corps K satisfaisant le théorème de l’axe principal à l’aide de propriétés des formes trace des extensions finies de K. Grâce à la caract érisation de ces mêmes corps due à Waterhouse, on retrouve à partir de là, de façon élémentaire, un résultat de Becker selon lequel un pro-2-groupe qui se réalise comme groupe de Galois absolu d’un tel corps K est engendré par des involutions.

Abstract

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 involutions.

Keywords

Primary: 11E10secondary: 12D15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 1 , 01 March 1997 , pp. 81 - 87
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Becker, Eberhard, Euklidische Körper und euklidische Hüllen von Körpern, Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, II., J. Reine Angew. Math. 268/269 (1974), 41–52.Google Scholar
2. Lam, Tsit Yuen, The Algebraic Theory of Quadratic Forms, Math. Lecture Note Ser., W. A. Benjamin, Inc., Reading, Massachusetts, 1973.Google Scholar
3. Waterhouse, William C., Self-adjoint operators and formally real fields, Duke Math. J. (2) 43 (1976), 237–243.Google Scholar