A ring homomorphism from $\mathbb{C}$ to $\mathbb{C}$

Question

For a nontrivial homomorphism from $\mathbb{C}$ to $\mathbb{C}$, I know it’s invective and maps $1$ to $1$ and i to +i or -i. Then is such a map onto? I can’t go further.

Answer 1

If you believe Zorn's lemma, there are lots of field maps from $\Bbb C$ to itself, not all surjective. Take a transcendence basis $A$ of $\Bbb C$ over $\Bbb Q$. Then $A$ has cardinality $|\Bbb C|$. Take an injection $\phi:A\to A$. Then, $\phi$ extends to a field map $\Phi$ from $\Bbb C$ to itself (this requires Zorn, as does the existence of $A$). If $\phi$ is not surjective, neither is $\Phi$.