What are the axioms that differentiate a complex field from a real field? Excluding one from the other.
Is it fair to say that:
a. A complex field is a field with the extra (?) axiom has an element $i$ such that $i^2 + 1 = 0$
b. A real field is a field where such element does not exists (?)
I am trying to find a definition that is mutually exclusive between a real and a complex field. However I have two problems:
It is weird to define a field (case b) by something that it doesn't have. Also it can't be that complex filed adds an axiom, if so the definition doesn't seem to be exclusive.
When defining the field of reals, the non existence of element $i$, is not mentioned (although I guess one can demonstrate by some means that $\forall a \neq 0, a^2 > 0$, which in turn is like saying $\nexists i / i^2 = -1$).
Finally, how does conjugation enters in the picture? can the existence of a conjugation itself belong to an axiom? For example $\forall z, \exists \bar z / z\cdot\bar z > 0$.