Problem:
Let $p$ be a prime and $\Bbb{Z}_p$ be the set of residue classes modulo $p$. Prove that no matter what proposed ordering is assigned to the finite field, one of the 4 ordering axioms will fail. Thus, it is not possible to order the field's elements in any way.
My solution:
Let $\overline{a} \in \Bbb{Z}_p$ and the residue class $\overline{a}$ corresponds to the value $a \in \Bbb{Z}:a>0$
The sum of two strictly positive elements should be strictly positive
Let $b = p - a$. Because $0<a<p$ we have $a<p \implies 0<p-a$
Thus: $\overline{a}+\overline{p-a} = (a+(p-a))\mod{p} = p\mod{p}=0$
Thus we have: $\overline{a}+\overline{b} = \overline{a}+\overline{p-a} = \overline{0}$ but $0 < a$
Thus, by contradiction, the set's elements cannot be ordered.
Is this right? Thanks :)