Group Theory – Prove Kernel of Homomorphism $f: \mathbb{C}^* \rightarrow \mathbb{R}^*$ is Infinite

Question

Let $f: \mathbb{C}^* \rightarrow \mathbb{R}^*$ be a homomorphism of the multiplicative group of complex numbers to the multiplicative group of real numbers. I need to show that the kernel of $f$ must be infinite.

I do know that $\mathbb{C}^*$ and $\mathbb{R}^*$ are not isomorphic to each other from here. So does that mean $f$ is not onto? But how will I be able to show that the kernel is infinite?

Thanks in advance.

Answer 1

Hint: There are infinitely many complex numbers $z$ that satisfy the equation $z^n=1$ for some odd integer $n$. What can you say about their homomorphic images?