## Theorem

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

## Proof

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

 $(V1)$ $:$ $\displaystyle \forall q, r \in \Q:$ $\displaystyle \nu_p \left({q r}\right)$ $\displaystyle =$ $\displaystyle \nu_p \left({q}\right) + \nu_p \left({r}\right)$ $(V2)$ $:$ $\displaystyle \forall q \in \Q:$ $\displaystyle \nu_p \left({q}\right) = +\infty$ $\displaystyle \iff$ $\displaystyle q = 0$ $(V3)$ $:$ $\displaystyle \forall q, r \in \Q:$ $\displaystyle \nu_p \left({q + r}\right)$ $\displaystyle \ge$ $\displaystyle \min \left\{ {\nu_p \left({q}\right), \nu_p \left({r}\right) }\right\}$

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

### Axiom $(V1)$

 $\displaystyle \nu_p \left({q r}\right)$ $=$ $\displaystyle \nu_p \left({\frac a b \cdot \frac c d}\right)$ $\displaystyle$ $=$ $\displaystyle \nu_p \left({\frac {a c} {b d} }\right)$ Definition of Rational Multiplication $\displaystyle$ $=$ $\displaystyle \nu_p^\Z \left({a c}\right) - \nu_p^\Z \left({b d}\right)$ Definition of $p$-adic Valuation $\displaystyle$ $=$ $\displaystyle \left({\nu_p^\Z \left({a}\right) + \nu_p^\Z \left({c}\right)}\right) - \left({\nu_p^\Z \left({b}\right) + \nu_p^\Z \left({d}\right)}\right)$ $p$-adic Valuation on Integers is Valuation: Axiom $V1$ $\displaystyle$ $=$ $\displaystyle \nu_p^\Z \left({a}\right) - \nu_p^\Z \left({b}\right) + \nu_p^\Z \left({c}\right) - \nu_p^\Z \left({d}\right)$ Integer Addition is Commutative $\displaystyle$ $=$ $\displaystyle \nu_p \left({\frac a b}\right) + \nu_p \left({\frac c d}\right)$ Definition of $p$-adic Valuation $\displaystyle$ $=$ $\displaystyle \nu_p \left({q}\right) + \nu_p \left({r}\right)$

$\Box$

### Axiom $(V2)$

 $\displaystyle \dfrac a b$ $=$ $\displaystyle 0$ $\displaystyle \iff \ \$ $\displaystyle a$ $=$ $\displaystyle 0$ Definition of Rational Number $\displaystyle \iff \ \$ $\displaystyle \nu_p^\Z \left({a}\right)$ $=$ $\displaystyle +\infty$ Definition of $p$-adic Valuation $\displaystyle \iff \ \$ $\displaystyle \nu_p^\Z \left({a}\right) - \nu_p^\Z \left({b}\right)$ $=$ $\displaystyle +\infty$ as $b \ne 0$ $\displaystyle \iff \ \$ $\displaystyle \nu_p \left({\frac a b}\right)$ $=$ $\displaystyle +\infty$ Definition of $p$-adic Valuation

$\Box$

### Axiom $(V3)$

From P-adic Valuation on Integers follows that:

 $\displaystyle \nu_p \left({\frac a b + \dfrac c d}\right)$ $=$ $\displaystyle \nu_p \left({\frac {a d + b c} {b d} }\right)$ Definition of Rational Addition $\displaystyle$ $=$ $\displaystyle \nu_p^\Z \left({a d + c b}\right) - \nu_p^\Z \left({b d}\right)$ Definition of $p$-adic Valuation $\displaystyle$ $\ge$ $\displaystyle \min \left\{ {\nu_p^\Z \left({a d}\right), \nu_p^\Z \left({c b}\right)}\right\} - \nu_p^\Z \left({b d}\right)$ $p$-adic Valuation on Integers is Valuation: Axiom $V3$ $\displaystyle$ $=$ $\displaystyle \min \left\{ {\nu_p^\Z \left({a}\right) + \nu_p^\Z \left({d}\right), \nu_p^\Z \left({c}\right) + \nu_p^\Z \left({b}\right)}\right\} - \nu_p^\Z \left({b}\right) - \nu_p^\Z \left({d}\right)$ $p$-adic Valuation on Integers is Valuation: Axiom $V1$ $\displaystyle$ $=$ $\displaystyle \min \left\{ {\nu_p^\Z \left({a}\right) - \nu_p^\Z \left({b}\right), \nu_p^\Z \left({c}\right) - \nu_p^\Z \left({d}\right)}\right\}$ $\displaystyle$ $=$ $\displaystyle \min \left\{ {\nu_p \left({\frac a b}\right), \nu_p \left({\frac c d}\right)}\right\}$ Definition of $p$-adic Valuation

Hence:

$\nu_p \left({\dfrac a b + \dfrac c d}\right) \ge \min \left\{ {\nu_p \left({\dfrac a b}\right), \nu_p \left({\dfrac c d}\right)}\right\}$

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

$\blacksquare$