Proving that a set is not an ordered field Problem:
Let $S=\left \{ 0,1,2 \right \}$. How can someone prove that there is unique way of defining addition and multiplication such that $S$ is a field if $0$ of the set $S$ has the meaning (for any element $a$ in $S$: $0+a=a$), and $1$ in $S$ has the meaning (for any element $a$ in $S$: $1.a=a$)? Also,  can $S$ be an ordered field? 

The axioms for a ring already imply that $0\cdot a=0$ for all $a\in S$, so you have almost all of the multiplication table:
$$\begin{array}{c|cc}
\cdot&0&1&2\\ \hline
0&0&0&0\\
1&0&1&2\\
2&0&2
\end{array}$$
And $2$ has to have a multiplicative inverse, so we must have $2\cdot 2=1$.
For addition we automatically have this much:
$$\begin{array}{c|cc}
+&0&1&2\\ \hline
0&0&1&2\\
1&1&&\\
2&2&
\end{array}$$
Now $1$ must have an additive inverse, so either $1+1=0$, or $1+2=0$. Suppose that $1+1=0$; then what is $1+2$? If $1+2=0$, then $1=1+0=1+(1+2)=(1+1)+2=0+2=2$, which is absurd. If $1+2=1$, then $0=1+1=1+(1+2)=(1+1)+2=0+2=2$, which is also absurd. Thus, $1+2=2$. But we know that $2$ has an additive inverse $-2$, even if we don’t yet know what it is, so $1=1+\big(1+(-2)\big)=(1+2)+(-2)=2+(-2)=0$, which is also impossible. Thus, $1+1$ cannot be $0$. It also can’t be $1$ (why not?), so it must be $2$, and we have
$$\begin{array}{c|cc}
+&0&1&2\\ \hline
0&0&1&2\\
1&1&2&\\
2&2&
\end{array}$$
Clearly $2$ must be $-1$, giving us
$$\begin{array}{c|cc}
+&0&1&2\\ \hline
0&0&1&2\\
1&1&2&0\\
2&2&0
\end{array}$$
and it’s not hard to check that $2+2$ can now only be $1$.
Added: By the way, that last bit can be shortened to practically nothing if you recall that up to isomorphism there is only one group of order $3$, $\Bbb Z/3\Bbb Z$: that gives you the addition table right away.
How to see that $S$ cannot be an ordered field depends on how you’ve defined ordered field. If you’ve defined it in terms of a positive cone, note that $1+1=2=-1$, so $1$ can’t be in the positive cone: it’s closed under addition and never contains a non-zero element and its additive inverse. Similarly, $(-1)+(-1)=2+2=1$, so $-1$ can’t be in the positive cone, either. But this is also impossible: one of them has to be in it.
If you’ve defined it in terms of an order relation, you can get essentially the same contradictions. If $0<1$, then $1=0+1<1+1=2$, $2=1+1<2+1=0$, so by transitivity $0<0$; OOPS! A similar problem arises if $1<0$.
