The ordering I'm trying to consider is simply for $x,y \in \mathbb{C}$ then $x=y$.
I'm going through Rudin's Principles of Mathematical Analysis and the only restrictions he gives for an ordered field are that it has some total order and follows a few more rules.
- For $x,y$ exactly one of the following holds: $x<y$, $x=y$, $x>y$.
- If $x<y$ and $y<z$ then $x<z$.
- If $y<z$ then $x+y < x+z$.
- If $x,y > 0$ then $xy > 0$.
As I see it, this trivial ordering satisfies 1 and vacuously satisfies 2-4. Thus $\mathbb{C}$ is ordered and ever field can be ordered likewise.
When I looked on Wikipedia, they note that since $-1$ is a square, it must be positive thus no ordering is possible. But I believe that should say $-1$ is nonnegative, and that all numbers or nonnegative. With all numbers equal to each other and $0$, there is no issue.
Am I missing something?
What you describe isn't a total order (in fact, abuse of notation aside, it's a preorder).