Let $\mathscr{F}=\{0,1\}$ and suppose that addition and multiplication are defined on $\mathscr{F}$ not in the usual way but by means of the "peculiar" addition and multiplication tables below:
$$\begin{array}{c|ccc}+&0&1\\\hline0&0&1\\1&1&0\end{array} \qquad \begin{array}{c|cc}\times&0&1\\\hline0&0&0\\1&0&1\end{array}$$
(Note that $1+1=0$.) Show, by considering all possible values for $a$, $b$, and $c$ that $\mathscr{F}$ is a field under the "peculiar" rules of addition and multiplication given.
Explain why no ordering of the field $\mathscr{F}$ is compatible with its arithmetic (i.e. all possible orderings violate one of the axioms of order)
I was able to prove that $\mathscr{F}$ is a domain, but how do I prove that it doesn't have an arithmetic compatible order, can you give me some help? thanks.