Is there any onto homomorphism from $\mathbb{C}^*$ to $\mathbb{R}^*$?

Question

$\mathbb{C}^*$ be the multiplicative group of nonzero complex numbers similarly $\mathbb{R}^*$. If so can you provide an example

Answer 1

It seems that no such homomorphism exists. Assume that $\varphi\colon \mathbb{C}^*\to \mathbb{R}^*$ is a homomorhism. Take any $z\in \mathbb{C}^*$ and any $w\in \mathbb{C}^*$ such that $w^2=z$. (Every complex number has a square root.) Then $\varphi(z)=\varphi(w^2)=\varphi(w)^2>0$. Thus $\varphi(z)$ is always positive.