Descent problem for certificate of non-negativity on semi-algebraic sets

Let $F$ be a subfield of $\mathbb{R}$ and let $K$ be a basic closed semi-algebraic set in $\mathbb{R}$ with $\mathrm{\partial }K\subset F$. Let $\mathcal{N}$ be the natural choice of generators of $K$. We show that if $f\in F[x]$ is $\ge 0$ on $K$, then $f$ can be written as $$f=\sum _{e\in \{0,1{\}}^s}{a}_e{\sigma }_e{g}^e$$ where $a_e\in F_{\ge 0}$, ${\sigma }_e\in \sum F[x{]}^2$ and $g^e=g_1^{e_1}\cdots g_s^{e_s}$. In other words, the preordering $T_{\mathcal{N}}$ of $F[x]$ is saturated. In case $F=\mathbb{R}$, this result is due to Kuhlmann and Marshall. As an application, we prove that if $K$ is compact, then $M_{\mathcal{N}}=T_{\mathcal{N}}=Pos(K)$. In other words, the quadratic module $M_{\mathcal{N}}$ of $F[x]$ is saturated.