Definition of field of fractions of $p$-adic integers

Question

I am given this definition of the field of fractions of the p-adic integers:

$$Q_p=\left\{ \frac{r}{s} \mid r, s \in Z_p, s \neq 0\right\}$$

How can I show that:

$Q_p$ consists of the sums of the form $\sum_{i=-k}^{\infty} a_ip^i$ ?

Answer 1

A $p$-adic integer can be expressed uniquely in the form $$ \sum_{i=0}^{\infty}a_{i}p^{i} $$ where $a_i$ is an integer with $0\le a_i<p$. Moreover, any $p$-adic integer can be written in a unique way as $$ up^k $$ with $u$ invertible and $k\ge0$. In particular, $$ \frac{r}{s}=\frac{r}{up^k}=\frac{u^{-1}r}{p^k} $$ Write $u^{-1}r=\sum_{i=0}^{\infty}a_{i}p^{i}$ and you're done.