Example of field which is ordered field which can be ordered in 2 distinct way

Question

I know that C is field which can be ordered in many ways like dictionary order etc but it is not ordered field .I wanted to find ordered field which can be ordered in distinct ways still remains ordered field with respect to that ordered.Any Help will be appreciated

Answer 1

How about $\Bbb Q[X]/\langle(X^2-2)\rangle$? We can order this either by decreeing $a+bX>0$ when $a+b\sqrt2>0$, or by decreeing that $a+bX>0$ when $a-b\sqrt2>0$.