Abstract

A chain-closed field is defined as a chainable field (i.e. a real field such that, for all n ∈ N, Σ K2n+1 ≠ Σ K2n) 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 ν K is odd divisible with |ν K/2ν K| = 2. If K admits only one such valuation, we show that f ∈ 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 = 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.