$n$-th roots of unity in general field

Question

It is well known that if $\mathbb{F}$ is a finite field, then the group $$ G=\{x\in \mathbb{F}:x^n=1\} $$ is cylic. Is it true also when $\mathbb{F}$ is infinite?

Answer 1

Yes, the group of $n$-th roots of unity is cyclic for all fields. We can use the following result:

Theorem: Any finite subgroup of the multiplicative group of any field is cyclic.

There are several references for this on MSE. Fabio gave one above.