Formally real field with two different orders

Question

If $F$ is a formally real field, then there exists a total order relation on $F$ which is compatible with its sum and product, but it needs not be unique.

a) How different can be two (compatible with the field) order relations over the same formally real field, in the sense of not sharing important properties (e.g. being Archimedean)? Can you give specific examples?

b) Can the interrelation between two different order relations over the same formally real field be used for proving any interesting result?

Answer 1

A simple example, $F = \mathbb Q(X)$. For any transcendental real number $\alpha$, we can order $F$ be letting $X=\alpha$. We can also order $F$ by letting $X$ exceed all elements of $\mathbb Q$. Or with $0 < X < r$ for all positive rationals $r$. There are others, too.

Answer 2

I can just make you an example. For the zeromello's lemma in every set is possibile to impose a good order (such that every subsets has minimal element) which is very different from the standard order on R. So i belive that the answer to point a is yes order can have really different propreties.