Prove that the field of complex numbers cannot be ordered.

An ordered field F is a field having a subset P satisfying the following:

1) For all x,y in P, x+y in P
2) For all x,y in P, x*y in P
3) For all x in F, exactly one of the statements

      (x in P,  x = 0,  -x in P) is true.

From the third constraint, we have that either i or -i must be member of P. If i is in P, then i*i = -1 is also in P, and i*(-1) must also be in P. The same goes for -i : (-i)*(-i) = -1, (-1)*(-i) = i. So either both i or -i are in P, or neither is.

Posted by Robby Goetschalckx on 2007-06-06 09:57:32