TITLE: A p-adic number is contained in $\mathbb Q$ iff it is periodic.
QUESTION [7 upvotes]: A p-adic number is contained in $\mathbb Q$ iff it is periodic.
I'm looking at the $\Rightarrow$ direction. I pretty much know the standard proof but I'm given a hint in my assignment which I can't see how it leads on to the rest of the proof.
Hint: Given $a\in\mathbb Q, a\neq0$. Write $a=p^0\frac{t}s$ with $p\nmid ts$ and show that $\exists r>0$ s.t. $1-p^r\equiv 0\pmod{s}$.
I don't even know if I'm correct in getting to the hint but here's what I have so far:
$p\nmid ts$ implies $p\nmid s$. By Euler's Theorem, $p^{\phi(s)}\equiv 1\pmod s$. So $r=\phi(s)>0$.
How can I carry on?

REPLY [6 votes]: Your work so far is correct. It seems like you're asking about how to conclude that $a$ is periodic in $p$-adic notation. Here is a hint:
Write $1-p^r=sk$ for some integer $k$, so that $a=\dfrac{tk}{1-p^r}$. Now, what is $\dfrac{1}{1-p^r}$ in $p$-adic notation?