# P-adic Valuation of Rational Number is Well Defined

## Theorem

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

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.

$\blacksquare$