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

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The discussion centers on demonstrating that elements of the field of fractions of the p-adic integers, Q_p, can be expressed as sums of the form ∑_{i=-k}^{∞} a_ip^i, where at least one index i is negative. It highlights that any element in Z_p can be represented as u · p^k, with u being a unit and k non-negative. The conversation points out that while Z_p is contained in Q_p, the statement holds true in Q_p excluding Z_p. The user aims to leverage this understanding to show that if x belongs to Q_p with |x|_p ≤ 1, then x must be in Z_p. The necessity of having at least one negative index i is emphasized for the argument to hold.
evinda
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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)
 
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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.
 
mathbalarka said:
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)
 
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