How to show if $x\in\mathbb{Q}$ ,then there exists $N \ge0$ such that $p^Nx\in\mathbb{Z}_p$?

Question

How to show if $x\in\mathbb{Q}$ ,then there exists $N \ge0$ such that $p^Nx\in\mathbb{Z}_p$? Or are there any references?

Answer 1

Let $q$ be a prime different from $p$. Here’s an explicit way of finding $1/q$ as a $p$-adically convergent series of ordinary integers:

Since $q\not\equiv0\pmod p$, you get $q^{p-1}\equiv1\pmod p$, in other words $q^{p-1}=1+mp$ for some ordinary integer $m$. Thus we have \begin{align} \frac1{q^{p-1}}&=1-mp+m^2p^2-\cdots=\sum_{i\ge0}(-mp)^i\\ \frac1q&=q^{p-2}\sum_{i\ge0}(-mp)^i\,, \end{align} where the infinite sums are convergent, the common ratio being $p$-adically smaller than $1$.

You see that primality of $q$ was not used here: the argument is valid for any integer not divisible by $p$.