## Journal of Symbolic Logic

### XVIIeme Probleme de Hilbert sur les Corps Chaine-Clos

#### Abstract

A chain-closed field is defined as a chainable field (i.e. a real field such that, for all $n \in \mathbf{N}, \mathbf{\Sigma K^{2n+1}} \neq \mathbf{\Sigma K^{2n}}$) which does not admit any "faithful" algebraic extension, and can also be seen as a field having a Henselian valuation $\nu$ such that the residue field $K/\nu$ is real closed and the value group $\nu K$ is odd divisible with $|\nu K/2\nu K| = 2$. If $K$ admits only one such valuation, we show that $f \in K(X)$ is in $\mathbf{\Sigma} K(X)^{2n} \operatorname{iff}$ for any real algebraic extension $L$ of $K, "f(L) \subseteq \mathbf{\Sigma}L^{2n}"$ holds. The conclusion is also true for $K = \mathbf{R}((t))$ (a chainable but not chain-closed field), and in the case $n = 1$ it holds for several variables and any real field $K$.

#### Article information

Source
J. Symbolic Logic, Volume 56, Issue 3 (1991), 853-861.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183743733

Digital Object Identifier
doi:10.2178/jsl/1183743733

Zentralblatt MATH identifier
0746.03026

JSTOR