Hostname: page-component-7bb8b95d7b-2h6rp Total loading time: 0 Render date: 2024-09-06T06:56:24.916Z Has data issue: false hasContentIssue false

XVIIème problème de Hilbert sur les corps chaîne-clos

Published online by Cambridge University Press:  12 March 2014

Françoise Delon et Danielle Gondard*
Affiliation:
Uer de Mathématiques, Université Paris-VII, 75251 Paris, France Departement de Mathématiques, Université Paris-VI, 75252 Paris, France

Abstract

A chain-closed field is defined as a chainable field (i.e. a real field such that, for all n ∈ N, ΣK 2n+2 ≠ ΣK 2n ) which does not admit any “faithful” algebraic extension, and can also be seen as a field having a Henselian valuation ν such that the residue field K/ν is real closed and the value group νK is odd divisible with ∣νK/2νK∣ = 2. If K admits only one such valuation, we show that fK(X) is in ΣK(X)2n for any real algebraic extension L of K,“f(L) ⊆ ΣL 2n ” holds. The conclusion is also true for K = R((t))(a chainable but not chain-closed field), and in the case n = 1 it holds for several variables and any real field K.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

RÉFÉRENCES

[B] Becker, R., Hereditarily-Pythagorean fields and orderings of higher types, Monografias de Matemática, vol. 29, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1978.Google Scholar
[BJ] Becker, E. and Jacob, B., Rational points on algebraic varieties over a general real closed field: a model theoretic approach, Journal für die Reine und Angewandte Mathematik, vol. 357 (1985), pp. 7795.Google Scholar
[D1] Delon, F., Quelques propriétés des corps valués en théorie des modè;les, Thèse de Doctorat d'État, Université Paris-VII, Paris, 1982.Google Scholar
[D2] Delon, F., Corps et anneaux de Rolle, Proceedings of the American Mathematical Society, vol. 97 (1986), pp. 315319.CrossRefGoogle Scholar
[D3] Delon, F., Compléments sur les corps chaîne-clos, Séminaire sur les structures algébriques ordonnées, Publications Mathématiques de l'Université Paris-VII, vol. 32, Paris, 1990, pp. 7994.Google Scholar
[Di] Dickmann, M., The model theory of chain closed fields, this Journal, vol. 53 (1988), pp. 921930.Google Scholar
[G1] Gondard, D., Théorie du premier ordre des corps chaînables et des corps chaîne-clos, Comptes Rendus des Séances de l'Académie des Sciences. Série I: Mathématique, vol. 304 (1987), pp. 463465.Google Scholar
[G2] Gondard, D., Chainable fields and real algebraic geometry, Real analytic and algebraic geometry: proceedings, Trento, 1988 (Galbiati, M. and Tognoli, A., editors), Lecture Notes in Mathematics, vol. 1420, Springer-Verlag, Berlin, 1990, pp. 128148.CrossRefGoogle Scholar
[H] Harman, J., Chains of higher level orderings, Ordered fields and real algebraic geometry (Dubois, D. W. and Recio, T., editors), Contemporary Mathematics, vol. 8, American Mathematical Society, Providence, Rhode Island, 1982, pp. 141174.CrossRefGoogle Scholar
[J] Jacob, B., The model theory of generalized real closed fields, Journal für die Reine und Angewandte Mathematik, vol. 323 (1981), pp. 213220.Google Scholar
[Rib] Ribenboim, P., Théorie des valuations, Les Presses de l'Université de Montréal, Montréal, 1964.Google Scholar
[RZ] Robinson, A. and Zakon, E., Elementary properties of ordered abelian groups, Transactions of the American Mathematical Society, vol. 96 (1960), pp. 222236.CrossRefGoogle Scholar
[S] Schmitt, P., Model theory of ordered abelian groups, Habilitationsschrift, Heidelberg, 1982.Google Scholar