0

I was wondering what are the ring homomorphism $f$ from $\Bbb R \to \Bbb C$ and to $\Bbb C \to \Bbb C$ ?

I think that there is only one ring homomorphism from $\Bbb R \to \Bbb C$ : the inclusion. I know that the only ring homomorphism from $\Bbb R \to \Bbb R$ is the identity and I feel that the arguments used to prove it also works for the case of ring homomorphism from $\Bbb R \to \Bbb C$.

For $\Bbb C \to \Bbb C$, the restriction to $\Bbb R$ is the inclusion if what I said previously is true. So maybe the identity is the only ring homomorphism from $\Bbb C \to \Bbb C$, but I struggle to show it and I feel that some complex analysis will be needed.

Kilkik
  • 1,899
  • 3
    I bet you know another ring homomorphism $\Bbb C\to\Bbb C$. Think back to when you first were taught complex numbers. What's the first thing you learn that you can do to any complex number that's truly a new concept that you never learned about for real numbers? Also, finding all the homomorphisms (of either type) gets into weird Axiom of Choice territory. How familiar are you with the AoC? – Arthur Jun 16 '22 at 18:39
  • 1
    For the ring homomorphisms $\Bbb C\rightarrow \Bbb C$ see this duplicate. – Dietrich Burde Jun 16 '22 at 18:45

0 Answers0