I am reading S. Roman's book "Field Theory". In chapter $1$ I found the following exercise
Let $F^*$ be the multiplicative group of all nonzero elements of a field $F$. We have seen that if G is a finite subgroup of $F^*$, then G is cyclic. Prove that if $F$ is an infinite field then no infinite subgroup $G$ of $F^*$ is cyclic.
I think that the question is wrong; for instance, if $F = \mathbb{Q}$, then the set $G = \{ 2^n : n \in \mathbb{Z}\}$ is an infinite cyclic subgroup of $\mathbb{Q}^*$
am I right or am I misunderstanding something?