Consider a “pseudo-ordering” on the complex field defined by saying a complex number is positive if and only if is positive. Which of the three order axioms (listed in the previous exercise) are satisfied.
The final axiom holds since is true for all . Hence, is not positive so the axiom holds.
The first axiom fails since under this ordering both and are positive, but is not positive.
The second axiom fails since for any complex number we have which implies that if is positive then so is .