I was just thinking that as rational functions form an ordered field you could describe...

The field $\mathbb{R}(x)$ of real rational functions is ordered by the condition that $\frac{r_o{x}^n+...+r_n}{s_o{x}^m+...+s_m}>0$ if $r_0,s_o>0$.
This gives rise to a topology, which is metrizable:The reason is that there is a denumerable set, consisting of the fractions $\frac{1}{x^N}$, which is cofinal in the sense that for every positive rational real function $\frac{p(x)}{q(x)}>0$, there exists $N$ with $0<\frac{1}{x^N}<\frac{P(x)}{q(x)}$.
This implies that the ordered field $\mathbb{R}(x)$ is metrizable by a theorem of Dobbs.
The whole paper is interesting and might serve as the reference you are looking for.