I read that the set of rational functions with rational coefficients forms an ordered field, yet it is non-archimedean. I tried googling this, but I don't think I understood the solution.
- How does one define an order on rational functions of the form $\mathbb{Q}(x)=p(x)/q(x)$?
- How do you show that $\mathbb{Q}(x)$ is non-archimedean? That there is no natural number $n$, such that $n>\mathbb{Q}(x)$? Do I substitute numerical values of $x$ and show that $\mathbb{Q}(x)$ is unbounded or something?