P-adic Valuation of Rational Number is Well Defined

Theorem
The $p$-adic valuation:


 * $\nu_p: \Q \to \Z \cup \set {+\infty}$

is well defined.

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

Thus:
 * $a d = b c \in \Z$

By :
 * $b, d \ne 0$

By :
 * $\map {\nu_p^\Z} b, \map {\nu_p^\Z} d < +\infty$

Case 1 : $a \ne 0$
Let $a \ne 0$.

It follows:


 * $c \ne 0$

By :


 * $\map {\nu_p^\Z} a, \map {\nu_p^\Z} c < +\infty$

Then:

Case 2 : $a = 0$
Let $a = 0$.

It follows:
 * $c = 0$

By :


 * $\map {\nu_p^\Z} a = \map {\nu_p^\Z} c = +\infty$

Then:

In either case:

So:

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