Integer Arbitrarily Close to Rational in Valuation Ring of P-adic Norm
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Theorem
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$.
Let $x \in \Q$ such that $\norm x_p \le 1$.
Then for all $i \in \N$ there exists $\alpha \in \Z$ such that:
- $\norm {x - \alpha}_p \le p^{-i}$
Proof
Let $i \in \N$.
Let $x = \dfrac a b: a, b \in \Z \text{ and } b \ne 0$.
Without loss of generality we can assume that $\dfrac a b$ is in canonical form.
By Valuation Ring of P-adic Norm on Rationals:
- $\dfrac a b \in \Z_{\paren p} = \set {\dfrac c d \in \Q : p \nmid d}$
So $p \nmid b$.
Since $p \nmid b$, by Prime not Divisor implies Coprime then $p^i \perp b$.
By Integer Combination of Coprime Integers, there exists $m, l \in \Z: m b + l p^i = 1$.
Then:
\(\ds \norm {\dfrac a b - am}_p\) | \(=\) | \(\ds \norm {\dfrac a b}_p \norm {1 - a m b}_p\) | Non-Archimedean Norm Axiom $\text N 2$: Multiplicativity | |||||||||||
\(\ds \) | \(\le\) | \(\ds \norm {1 - m b}_p\) | As $\norm {\dfrac a b}_p \le 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \norm {l p^i}_p\) | As $1 - m b = l p^i$ | |||||||||||
\(\ds \) | \(\le\) | \(\ds p^{-i}\) | Definition of $p$-adic norm on integers |
As $am \in \Z$, the result follows with $\alpha = am$.
$\blacksquare$
Also see
Sources
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.4$ The field of $p$-adic numbers $\Q_p$: Lemma $1.29$