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
.