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**particlejohn** Prove that there is no set $\displaystyle P \subset \mathbb{Z}_{3} $ (integers modulo $\displaystyle 3 $) which makes $\displaystyle \mathbb{Z}_{3} $ into an ordered field.

Assume for contradiction that there is a set $\displaystyle P \subset \mathbb{Z}_{3} $ which makes $\displaystyle \mathbb{Z}_{3} $ into an ordered field. Then there is a set $\displaystyle P \subset \mathbb{Z}_{3} $ such that if $\displaystyle a,b \in P, \ \text{then} \ a+b, ab \in P $ and if $\displaystyle a \in \mathbb{Z}_{3} $, then exactly one of the following is true: $\displaystyle a \in P, -a \in P, a = 0 $.

How would I obtain a contradiction. Is this the right way to go about proving it?