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XVIIème problème de Hilbert sur les corps chaîne-clos

Published online by Cambridge University Press:  12 March 2014

Françoise Delon et Danielle Gondard
Françoise Delon et Danielle Gondard*
Affiliation:
Uer de Mathématiques, Université Paris-VII, 75251 Paris, France Departement de Mathématiques, Université Paris-VI, 75252 Paris, France
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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 f ∈ K(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
Information
The Journal of Symbolic Logic , Volume 56 , Issue 3 , September 1991 , pp. 853 - 861
Copyright
Copyright © Association for Symbolic Logic 1991

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References

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