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**redsoxfan325** The complex numbers are not an ordered field.

Proof. Assume it is an ordered field. Then either $\displaystyle i>0$ or $\displaystyle i<0$ (because we know $\displaystyle i\neq0$). If $\displaystyle i>0$ then $\displaystyle i^2>0^2 \implies -1>0$, which is clearly false, so $\displaystyle i\not>0$. If $\displaystyle i<0$, then $\displaystyle -i>0$ and $\displaystyle (-i)^2>0^2 \implies -1>0$, which is clearly false, so $\displaystyle i\not<0$. Thus, none of the following are true: $\displaystyle i>0, i<0$, or $\displaystyle i=0$. Thus $\displaystyle \mathbb{C}$ is not an ordered field.