Theorem:If F is an ordered field, then the square of any non-zero element is positive.

Proof:Either x>0 or x<0, by trichtonomytheorem. If x>0 then x*x=x^2>0 by closure property.

If x<0 then -x>0 trichtonomypropertybut then, (-x)*(-x)=x^2>0 by closure property.

Q.E.D.

Corollary:In an ordered field (note it cannot be trivial). 1>0

Proof:

1=1^2 use above result.

Q.E.D.

----

Note, this prooves the complex numbers cannot be ordered because 1^2>0 bet yet i^2=-1<0.

Contradiction.