Integers are Arbitrarily Close to P-adic Integers
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Theorem
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
Let $x \in \Z_p$.
Then for $n \in \N$ there exists unique $\alpha \in \Z$:
- $(1): \quad 0 \le \alpha \le p^n - 1$
- $(2): \quad \norm { x -\alpha}_p \le p^{-n}$
Proof
Let $n \in \N$.
From Rational Numbers are Dense Subfield of P-adic Numbers:
- the rational numbers are dense in $\Q_p$.
So there exists:
- $\dfrac a b \in \Q: \norm {x - \dfrac a b}_p \le p^{-n}$
From Unique Integer Close to Rational in Valuation Ring of P-adic Norm, there exists unique $\alpha \in \Z$ such that:
- $\norm {\dfrac a b - \alpha}_p \le p^{-n}$
- $0 \le \alpha \le p^n - 1$
Then:
\(\ds \norm {x - \alpha}_p\) | \(=\) | \(\ds \norm {\paren {x - \dfrac a b} + \paren {\dfrac a b - \alpha} }_p\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \max \set {\norm {x - \dfrac a b}_p, \: \norm {\dfrac a b - \alpha}_p }\) | Norm axiom (N4) (Ultrametric Inequality) | |||||||||||
\(\ds \) | \(\le\) | \(\ds p^{-n}\) |
Now suppose $\beta \in \Z$ also satisfies conditions $(1)$ and $(2)$, that is:
- $0 \le \beta \le p^n - 1$
- $\norm { x - \beta}_p \le p^{-n}$
Then:
\(\ds \norm{\dfrac a b - \beta}_p\) | \(=\) | \(\ds \norm{\paren{\dfrac a b - x} + \paren{x - \beta} }_p\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \max \set{\norm{\dfrac a b - x}_p,\:\norm{x - \beta}_p}\) | Norm axiom (N4) (Ultrametric Inequality) | |||||||||||
\(\ds \) | \(\le\) | \(\ds \max \set{\norm{x - \dfrac a b}_p,\:\norm{x - \beta}_p}\) | Norm of Negative | |||||||||||
\(\ds \) | \(\le\) | \(\ds p^{-n}\) |
From Unique Integer Close to Rational in Valuation Ring of P-adic Norm, $\alpha \in \Z$ was unique, so:
- $\beta = \alpha$
The result follows.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$: Lemma $3.3.4 \ \text{(ii)}$