P-adic Valuation is Valuation

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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.

$\blacksquare$


Sources