Complexes and Reals: The Impossibility of an Onto Ring Homomorphism

[MathMike91]

Onto ring homomorphism C-->R

Why can't there be an onto ring homomorphism from the Complexes to the Reals?

The only property of the complexes that the rations don't have that I can think of is the guarantee of square roots- but I can't see how that would interfere with an onto function.

 

[micromass]

Let f be an onto homomorphism. Let z be such that f(z)=-1. Consider a such that a²=z, then f(a)²=f(z)=-1. This can't happen...