P-adic Valuation is Valuation

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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.

$\blacksquare$


Sources

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