Characterization of Rational P-adic Unit

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

Let $\Z^\times_p$ be the $p$-adic units.

Let $\Q$ be the rational numbers.

Then:
 * $\Z^\times_p \cap \Q = \set{\dfrac a b \in \Q : p \nmid ab}$

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

We have: