Is the set $ \ (\mathbb{Z}_p \setminus \mathbb{Z}) \cap \mathbb{Q}$ non-empty? $\text{p-adic numbers}:$ **My questions are** - $(1)$ Is the set $ \ (\mathbb{Z}_p \setminus \mathbb{Z}) \cap \mathbb{Q}$ non-empty? $(2)$ Is the set $ \ (\mathbb{Z}_p \setminus \mathbb{Z}) \cap \mathbb{Q}_p $ non-empty? $(3)$ If non-empty , then what are the intersection sets ? I can not conclude the answer. Please someone help me with details answer or at least hints.

If $q$ is a prime distinct from $p$, then $1/q\in\mathbb{Z}_{p}\cap\mathbb{Q}$ and $1/q\
otin\mathbb{Z}$. Thus $1/q$ belongs to the set in $(1)$.

Since $\mathbb{Z}_p\setminus\mathbb{Z}\subseteq\mathbb{Q}_{p}$, the set in $(2)$ is the same as $\mathbb{Z}_p\setminus\mathbb{Z}$.