MHB How do we know that i is at least one negative?

[evinda]

Hi! (Smile)

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 $\sum_{i=-k}^{\infty} a_ip^i$, where $i$ takes at least one negative value ? (Thinking)

 

[mathbalarka]

Recall that any element in $\mathbf{Z}_p$ can be expressed as $u \cdot p^k$ for $k \geq 0$ and $u$ a unit of the ring $\mathbf{Z}_p$. So arbitrary elements in $\text{frac} \, \mathbf{Z}_p$ essentially looks like $m \cdot n^{-1} = m \cdot u^{-1} \cdot p^{-k} = a \cdot p^{-k}$ where $a \in \mathbf{Z}_p$ as $u$ is a unit. Thus, $\text{frac} \, \mathbf{Z}_p$ is really the field $S^{-1} \cdot \mathbf{Z}_p$ where $S = \{p^n : n \in \mathbb{N}\}$. How does the elements look like in here?

where $i$ takes at least one negative value?

Untrue, as $\mathbf{Z}_p$ sits inside $\mathbf{Q}_p$. It is true in $\mathbf{Q}_p \setminus \mathbf{Z}_p$ however.

 

[evinda]

Untrue, as $\mathbf{Z}_p$ sits inside $\mathbf{Q}_p$. It is true in $\mathbf{Q}_p \setminus \mathbf{Z}_p$ however.

I wanted to use this fact, in order to show that if $x \in \{ x \in Q_p | |x|_p \leq 1\}$, then $x \in Z_p$.

But, if $i$ does not get at least one negative value, it does not stand. (Worried)