P-adic Valuation of Rational Number is Well Defined

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Theorem

The $p$-adic valuation:

$\nu_p: \Q \to \Z \cup \left\{{+\infty}\right\}$

is well defined.


Proof

Let $\dfrac a b = \dfrac c d \in \Q$.

Thus:

$a d = b c \in \Z$

Then:

\(\displaystyle \nu_p^\Z \left({a}\right) + \nu_p^\Z \left({d}\right)\) \(=\) \(\displaystyle \nu_p^\Z \left({a d}\right)\) P-adic Valuation on Integers is Valuation: Axiom $V1$
\(\displaystyle \) \(=\) \(\displaystyle \nu_p^\Z \left({b c}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \nu_p^\Z \left({c}\right) + \nu_p^\Z \left({b}\right)\) P-adic Valuation on Integers is Valuation: Axiom $V1$
\((1):\quad\) \(\displaystyle \leadsto \ \ \) \(\displaystyle \nu_p^\Z \left({a}\right) - \nu_p^\Z \left({b}\right)\) \(=\) \(\displaystyle \nu_p^\Z \left({c}\right) - \nu_p^\Z \left({d}\right)\)


So:

\(\displaystyle \nu_p^\Q \left({\frac a b}\right)\) \(=\) \(\displaystyle \nu_p^\Z \left({a}\right) - \nu_p^\Z \left({b}\right)\) Definition of $p$-adic Valuation
\(\displaystyle \) \(=\) \(\displaystyle \nu_p^\Z \left({c}\right) - \nu_p^\Z \left({d}\right)\) from $(1)$
\(\displaystyle \) \(=\) \(\displaystyle \nu_p^\Q \left({\dfrac c d}\right)\) Definition of $p$-adic Valuation


Thus, by definition, $\nu_p: \Q \to \Z \cup \left\{{+\infty}\right\}$ is well defined.

$\blacksquare$