Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Sufficient Condition

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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:

$x$ be a rational number

Proof

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


Lemma 6

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

$\Box$


To show that $x$ is a rational number it is sufficient to show that $y$ is a rational number.


Let

$\ldots d_{k - 1} \ldots d_1 d_0 d_{k - 1} \ldots d_1 d_0 d_{k - 1} \ldots d_1 d_0$

be the periodic canonical expansion of $y$.

By definition of a canonical expansion:

$y = d_0 + d_1 p + \ldots + d_{k-1} p^{k-1} + d_0 p^k + d_1 p^{k+1} + \ldots + d_{k-1} p^{2k-1} + d_0 p^{2k} + \ldots$


Let $a = d_0 + d_1 p + \ldots + d_{k-1} p^{k-1}$.

Then:

$y = a \paren { 1 + p^k + p^{2k} + \ldots }$


Lemma 7

$1 + p^k + p^{2k} + p^{3k} + \ldots = \dfrac 1 {1 - p^k}$

$\Box$


Then:

$y = \dfrac a {1 - p^k}$

Hence:

$y$ is a rational number


It follows that $x$ is a rational number.

$\blacksquare$