Valuation Ideal of P-adic Norm on Rationals

Theorem
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.

The induced valuation ideal on $\struct {\Q,\norm {\,\cdot\,}_p}$ is the set:
 * $\mathcal P = p\Z_{(p)} = \set{ \dfrac a b \in \Q : p \nmid b, p \divides a}$

where $\Z_{(p)}$ is the induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$

Proof
Let $\nu_p: \Q \to \Z \cup \set {+\infty}$ be the $p$-adic valuation on $\Q$.

Then:

Let $\dfrac a b \in \Q$ be in canonical form.

Then $a \perp b$

Suppose $p \divides a$.

Then $p \nmid b$.

Hence:
 * $\map {\nu_p} {a} \gt 0 = \map {\nu_p} {b}$

Suppose $p \nmid a$.

Then:
 * $\map {\nu_p} {b} \ge 0 = \map {\nu_p} {a}$

So:
 * $\map {\nu_p} {a} \gt \map {\nu_p} {b}$ $p \nmid b$ and $p \divides a$

Hence:
 * $\mathcal P = \set{\dfrac a b \in \Q : p \nmid b, p \divides a }$

So:

Hence:
 * $\mathcal P = p\Z_{(p)}$