P-adic Valuation is Valuation

Theorem
The $p$-adic valuation $\nu_p: \Q \to \Z \cup \left\{{+\infty}\right\}$ is a valuation on $\Q$.

Proof
Let $\dfrac{a}{b}, \dfrac{c}{d}\in\Q$.

Verifying the requirements for a valuation in turn:

$(1)$
Verifying by explicit computation:

$(2)$
Observe that:

$(3)$
From P-adic Valuation on Integers follows that:


 * $\nu_p\left(\dfrac{a}{b}+\dfrac{c}{d}\right)=\nu_p\left(\dfrac{ad+bc}{bd}\right)=\nu_p^\Z(ad+cb)-\nu_p^\Z(bd)\geq\min\{\nu_p^\Z(ad),\nu_p^\Z(cb)\}-\nu_p^\Z(b)-\nu_p^\Z(d)=\min\{\nu_p^\Z(a)-\nu_p^\Z(b),\nu_p^\Z(c)-\nu_p^\Z(d)\}=\min\left\{\nu_p\left(\dfrac{a}{b}\right),\nu_p\left(\dfrac{c}{d}\right)\right\}$

Thus $\nu_p:\Q\to\Z\cup\{+\infty\}$ is a valuation on $\Q$ by definition.