Prove that there is no way to choose an ordering in C such that it becomes an ordered field.

I know that it is impossible because the remarks in my book say the group of complex numbers cannot become an ordered field, but I have no idea how to prove it.

Question

Prove that there is no way to choose an ordering in C such that it becomes an ordered field.

I know that it is impossible because the remarks in my book say the group of complex numbers cannot become an ordered field, but I have no idea how to prove it.

anonymous · Answer

To prove that a field is ordered, you have to satisfy two conditions:
$$\begin{cases} a + c \le b + c ~~\text{ if }~~ a \le b\
0 \le a ~~\text{ and }~~ 0 \le b \implies 0 \le a b\end{cases}$$

Try a proof by contradiction. Assume $i\le0$ and $i\ge0$ and see where that takes you.

anonymous · Answer

great! thank you! that will definitely get me started!