Is that the fact that the only possible finite (multiplicative) subgroup of the field $\mathbb{C}$ can only be those which is formed by root

Question

I have just learnt a theorem that says:

For the group of invertible elements $F^*$ of a field $F$, its finite subgroup is cyclic.

And an example is that for $F=\mathbb{C}$, we have the group generated by n-th roots of unity finite subgroups, and hence cyclic.

So I am now considering if the only finite subgroup of $\Bbb C^*$ are group generated by the n-th root of unity. If so, how may I prove that there is no other finite subgroup? If not, may I please ask for some other example? Thanks!

Answer 1

Hint:

If $\;C\;$ is a finite subgroup of $\;\Bbb C^*\;$ , then there exists

$$n\in\Bbb N\;\;\;s.t.\;\;\;\text{for all}\;\;\;c\in C\;,\;\;c^n=1$$