I don't know how to write in good English, so I will follow Hungerford's word from his book Algebra.
The following relation on the additive group $\mathbb{Q}$ of rational numbers is a congruence relation $a\sim b \leftrightarrow a-b\in \mathbb{Z}$. Denote for $\mathbb{Q}/\mathbb{Z}$ the set of all those equivalence classes. Let $p$ be a prime and $\mathbb{Z}(p^{\infty})=\{\overline{a/b}\in \mathbb{Q}/\mathbb{Z}\;|\; a,b \in \mathbb{Z}\text{ and }b=p^{i}\text{ for some } i\geq 0\}.$
I am trying to define a vector space structure on $\mathbb{Z}(p^{\infty})$ over a field $\mathbb{F}$, but I think it is impossible. For example, if $\mathbb{F}$ is uncountable, it can't be done. But I don't know to answer this question when $\mathbb{F}=\mathbb{Q}$ or $\mathrm{char}(\mathbb{F})=q$, where $q$ is a prime number.
Thanks for your help!