I was reading a book and, at some point, the author states that, for $A$ torsion-free abelian group (i.e $\Bbb{Z}$-module), the ranks of $\Bbb{Q}\otimes A$ and $A$ are the same.
I was able to prove this fact using $rank(\Bbb{Q}\otimes A)=rank(\Bbb{Q})rank(A)$ and $rank(\Bbb{Q})=1$, where the rank is, by definition, the cardinality of a maximal $\Bbb{Z}$-linearly independent system of a $\Bbb{Z}$-module.
More generally, I know that $\Bbb{Q}$ is a subgroup (i.e. $\Bbb{Z}$-submodule) of the group $\Bbb{Q}_p$ of $p$-adic numbers. So my questions are:
(1) what's the rank of $\Bbb{Q}_p$ over $\Bbb{Z}$?
(2) and what is the rank of $\Bbb{Q}_p$ over $\Bbb{Z}_p$?
Heuristics suggests to me that they are both $1$, but I failed to prove that.
The definition I know is the following: a $p$-adic number is an element of the form $x=p^{-m}\sum_{i=0}^\infty a_ip^i$, while a $p$-adic integers has the form $y=\sum_{i=0}^\infty a_ip^i$, where $m\in\Bbb{Z}$, $a_i\in \{0,\dots,p-1\}$. They main problem I have is that I don't know how to deal with "tails" like $\sum_{i=N}^\infty a_ip^i$ for some $N>0$.
May you help me, please?