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- Total ordering on complex numbers 3 answers
An ordered field is a field $F$ which is also an ordered set, such that:
(i) $x+y < x+z$ if $x$,$y$,$z \in F$ and $y<z$
(ii) $xy > 0$ if $x \in F$, $y \in F$, $x > 0$, and $y>0$
Prove that no order can be defined in the complex field that turns it into an ordered field. Hint: -1 is a square
To prove that there can be no order defined in the complex field, I used the second point of the definition.
Let $x=(0,1)$ and $y=(0,1)$. I state that $x>0$ and $y>0$ (but I am unsure if I can do that). Thus $xy$ should be strictly positive, yet $xy= i^2=-1 <0$, thus this disproves the fact that it is possible to define an order in the complex field.
Would this be an acceptable proof? Can I really state that $x=(0,1)>0$?