I know you are supposed to prove that 0 < 1 by contradiction, but i'm not sure how to do it exactly using only the ordered field axioms.
Proof: Either x>0 or x<0, by trichtonomy theorem. If x>0 then x*x=x^2>0 by closure property.
If x<0 then -x>0 trichtonomy property but then, (-x)*(-x)=x^2>0 by closure property.
Corollary:In an ordered field (note it cannot be trivial). 1>0
1=1^2 use above result.
Note, this prooves the complex numbers cannot be ordered because 1^2>0 bet yet i^2=-1<0.
1 > 0
Assume that the above theorem is false. We know that 1 cannot be equal to 0 and by trichotomy, the only option left is : 1 < 0.
If that is indeed true, then -1 > 0. Then if we square the expression
(-1)^2 > 0 and we will get 1 > 0. This contradicts the assumption that 0 < 1.
Hope that helps.