Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 6
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 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \begin{cases}
\ds 0 + \paren {\sum_{i \mathop = n}^\infty d_i p^i } & : -m = n \\ \ds \paren {\sum_{i \mathop = -m}^{n - 1} d_i p^i} + \paren {\sum_{i \mathop = n}^\infty d_i p^i } & : -m < n \end{cases}\) |
as $-m \le n$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \begin{cases}
\ds 0 + p^n \paren {\sum_{i \mathop = n}^\infty d_i p^{i - n} } & : -m = n \\ \ds \paren {\sum_{i \mathop = -m}^{n - 1} d_i p^i} + p^n \paren {\sum_{i \mathop = n}^\infty d_i p^{i - n} } & : -m < n \end{cases}\) |
extracting common $p^n$ factor from each term |
Let:
- $\ds r = \begin{cases}
0 & : -m = n \\ \ds \sum_{i \mathop = -m}^{n - 1} d_i p^i & : -m < n \end{cases}$
- $\ds y = \sum_{i \mathop = n}^\infty d_i p^{i - n}$
Then:
- $x = r + p^n y$
We have by hypothesisthat:
- $r \in \Q$
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
$\blacksquare$