Theorem

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

Proof

To prove that $\nu_p$ is a valuation it is necessary to demonstrate:

 $(\text V 1)$ $:$ $\ds \forall q, r \in \Q:$ $\ds \map {\nu_p} {q r}$ $\ds =$ $\ds \map {\nu_p} q + \map {\nu_p} r$ $(\text V 2)$ $:$ $\ds \forall q \in \Q:$ $\ds \map {\nu_p} q = +\infty$ $\ds \iff$ $\ds q = 0$ $(\text V 3)$ $:$ $\ds \forall q, r \in \Q:$ $\ds \map {\nu_p} {q + r}$ $\ds \ge$ $\ds \min \set {\map {\nu_p} q, \map {\nu_p} r}$

Let $q := \dfrac a b, r := \dfrac c d \in \Q$.

Axiom $(\text V 1)$

 $\ds \map {\nu_p} {q r}$ $=$ $\ds \map {\nu_p} {\frac a b \cdot \frac c d}$ $\ds$ $=$ $\ds \map {\nu_p} {\frac {a c} {b d} }$ Definition of Rational Multiplication $\ds$ $=$ $\ds \map {\nu_p^\Z} {a c} - \map {\nu_p^\Z} {b d}$ Definition of $p$-adic Valuation on Rational Numbers $\ds$ $=$ $\ds \paren {\map {\nu_p^\Z} a + \map {\nu_p^\Z} c} - \paren {\map {\nu_p^\Z} b + \map {\nu_p^\Z} d}$ Restricted $p$-adic Valuation is Valuation: Axiom $\text V 1$ $\ds$ $=$ $\ds \map {\nu_p^\Z} a - \map {\nu_p^\Z} b + \map {\nu_p^\Z} c - \map {\nu_p^\Z} d$ Integer Addition is Commutative $\ds$ $=$ $\ds \map {\nu_p} {\frac a b} + \map {\nu_p} {\frac c d}$ Definition of $p$-adic Valuation on Rational Numbers $\ds$ $=$ $\ds \map {\nu_p} q + \map {\nu_p} r$

$\Box$

Axiom $(\text V 2)$

 $\ds \dfrac a b$ $=$ $\ds 0$ $\ds \leadstoandfrom \ \$ $\ds a$ $=$ $\ds 0$ Definition of Rational Number $\ds \leadstoandfrom \ \$ $\ds \map {\nu_p^\Z} a$ $=$ $\ds +\infty$ Definition of Restricted $p$-adic Valuation $\ds \leadstoandfrom \ \$ $\ds \map {\nu_p^\Z} a - \map {\nu_p^\Z} b$ $=$ $\ds +\infty$ as $b \ne 0$ $\ds \leadstoandfrom \ \$ $\ds \map {\nu_p} {\frac a b}$ $=$ $\ds +\infty$ Definition of $p$-adic Valuation on Rational Numbers

$\Box$

Axiom $(\text V 3)$

From Restricted P-adic Valuation is Valuation follows that:

 $\ds \map {\nu_p} {\frac a b + \dfrac c d}$ $=$ $\ds \map {\nu_p} {\frac {a d + b c} {b d} }$ Definition of Rational Addition $\ds$ $=$ $\ds \map {\nu_p^\Z} {a d + c b} - \map {\nu_p^\Z} {b d}$ Definition of $p$-adic Valuation on Rational Numbers $\ds$ $\ge$ $\ds \min \set {\map {\nu_p^\Z} {a d}, \map {\nu_p^\Z} {c b} } - \map {\nu_p^\Z} {b d}$ Restricted $p$-adic Valuation is Valuation: Axiom $\text V 3$ $\ds$ $=$ $\ds \min \set {\map {\nu_p^\Z} a + \map {\nu_p^\Z} d, \map {\nu_p^\Z} c + \map {\nu_p^\Z} b} - \map {\nu_p^\Z} b - \map {\nu_p^\Z} d$ Restricted $p$-adic Valuation is Valuation: Axiom $\text V 1$ $\ds$ $=$ $\ds \min \set {\map {\nu_p^\Z} a - \map {\nu_p^\Z} b, \map {\nu_p^\Z} c - \map {\nu_p^\Z} d}$ $\ds$ $=$ $\ds \min \set {\map {\nu_p} {\frac a b}, \map {\nu_p} {\frac c d} }$ Definition of $p$-adic Valuation on Rational Numbers

Hence:

$\map {\nu_p} {\dfrac a b + \dfrac c d} \ge \min \set {\map {\nu_p} {\dfrac a b}, \map {\nu_p} {\dfrac c d} }$

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

$\blacksquare$