Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 6

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Theorem

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.

Let $x \in \Q_p$.

Let the canonical expansion of $x$ be eventually periodic.


Then:

$\exists r \in \Q, n \in \Z, y \in \Z_p$:
$(1) \quad x = r + p^n y$
$(2) \quad$the canonical expansion of $y$ is periodic


Proof

Let $\ldots d_i \ldots d_2 d_1 d_0 . d_{-1} d_{-2} \ldots d_{-m}$ be the canonical expansion of $x$.


By definition of eventually periodic there exists a finite sequence of $k$ digits of $x$:

$\tuple {d_{n + k - 1} \ldots d_{n+1} d_n }$

such that $n \ge 0$ and for all $s \in \Z_{\ge 0}$ and for all $j \in \set {0, 2, \ldots, k - 1}$:

$d_{n + j + s k} = d_{n + j}$

where $k$ is the smallest $k$ to have this property.


We have:

\(\ds x\) \(=\) \(\ds \sum_{i \mathop = -m}^\infty d_i p^i\) Definition of Canonical P-adic Expansion
\(\ds \) \(=\) \(\ds \begin{cases} 0 + \paren{\sum_{i \mathop = n}^\infty d_i p^i } & : -m = n \\ \paren{\sum_{i \mathop = -m}^{n - 1} d_i p^i} + \paren{\sum_{i \mathop = n}^\infty d_i p^i } & : -m < n \end{cases}\) as $-m \le n$
\(\ds \) \(=\) \(\ds \begin{cases} 0 + p^n \paren{\sum_{i \mathop = n}^\infty d_i p^{i - n} } & : -m = n \\ \paren{\sum_{i \mathop = -m}^{n - 1} d_i p^i} + p^n \paren{\sum_{i \mathop = n}^\infty d_i p^{i - n} } & : -m < n \end{cases}\) Extract common $p^n$ factor from each term


Let:

$\ds r = \begin{cases} 0 & : -m = n\\ \sum_{i \mathop = -m}^{n - 1} d_i p^i & : -m < n \end{cases}$
$\ds y = \sum_{i \mathop = n}^\infty d_i p^{i - n}$

Then:

$x = r + p^n y$


By definition $r \in \Q$.


Re-indexing the series for $y$, we have:

$\ds y = \sum_{i \mathop = 0}^\infty d_{i+n} p^i$

By definition of $p$-adic integer:

$y \in \Z_p$


By definition of canonical expansion, the canonical expansion of $y$ is:

$\ldots d_{n+i} \ldots d_{n+2} d_{n+1} d_n$

Recall that for all $s \in \Z_{\ge 0}$ and for all $j \in \set {0, 2, \ldots, k - 1}$:

$d_{n + j + s k} = d_{n + j}$

By definition of periodic:

the canonical expansion of $y$ is periodic

$\blacksquare$