The complex numbers cannot be made into an ordered field. There certainly exist "natural" total orders of the complex numbers (e.g. one can take a one to one correspondence between them and the real numbers, which are ordered, and copy that order; or one can take a lexicographic order on x+iy by mapping it to (x,y) or to (y,x)). But none of them respect the algebraic properties we'd want; the order doesn't "play nicely" with the field axioms.


Suppose P were the positive half of C, the complex numbers. Then i≠0, so either i∈P or -i∈P.

In either case, we have some j (either i or -i) in P such that j2=-1. But this is impossible for a positive element.