Characterization of Rational P-adic Integer

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Theorem

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

Let $\Z_p$ be the $p$-adic integers for some prime $p$.

Let $\Q$ be the rational numbers.


Then:

$\Z_p \cap \Q = \set{\dfrac a b \in \Q : p \nmid b}$

Proof

Let $\norm{\,\cdot\,}^\Q _p$ denote the $p$-adic norm on the rational numbers.

We have:

\(\ds \Z_p \cap \Q\) \(=\) \(\ds \set{\dfrac a b \in \Q : \norm {\dfrac a b}_p \le 1}\) Definition of $p$-adic integers
\(\ds \) \(=\) \(\ds \set{\dfrac a b \in \Q : \norm{\dfrac a b}^\Q_p \le 1}\) Rational Numbers are Dense Subfield of P-adic Numbers
\(\ds \) \(=\) \(\ds \set{\dfrac a b \in \Q : p \nmid b}\) Valuation Ring of P-adic Norm on Rationals

$\blacksquare$

Sources

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