Show that $Q_p / \mathbb Z$ is divisible as $\mathbb Z$-module.

Question

Let $p \in \mathbb N$ be a prime. Let

$$Q_p : = \left \{ x \in \mathbb Q : (\exists k \in \mathbb Z)\ \mathrm {and}\ (\exists n \in \mathbb N)\ \mathrm {such}\ \mathrm {that}\ x= \frac {k} {p^n} \right \}.$$

Show that $Q_p / \mathbb Z$ is divisible as $\mathbb Z$-module.

How should I proceed? Please help me.

Thank you in advance.

Answer 1

You need to prove that for $x=a/p^r$ and $n\in\Bbb N$ then there is a $y\in\Bbb Q_p$ with $ny\equiv x\pmod{\Bbb Z}$. You can reduce to the case where $n$ is prime. If $n=p$ that's easy: take $y=x/p$.

So, let $n\ne p$ also be a prime. Try $y=b/p^r$ with $b$ an integer. Then one needs $nb\equiv a\pmod{p^r}$. This congruence is soluble.