P-adic Valuation on Integers is Valuation

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Theorem

Let $\nu_p^\Z: \Z \to \Z \cup \left\{ {+\infty}\right\}$ be the $p$-adic valuation restricted to the integers.


Then $\nu_p^\Z$ is a valuation.


Proof

To prove that $\nu_p^\Z$ is a valuation it is necessary to demonstrate:

\((V1)\)   $:$     \(\displaystyle \forall m, n \in \Z:\)    \(\displaystyle \nu_p^\Z \left({m n}\right) \)   \(\displaystyle = \)   \(\displaystyle \nu_p^\Z \left({m}\right) + \nu_p^\Z \left({n}\right) \)             
\((V2)\)   $:$     \(\displaystyle \forall n \in \Z:\)    \(\displaystyle \nu_p^\Z \left({n}\right) = +\infty \)   \(\displaystyle \iff \)   \(\displaystyle n = 0 \)             
\((V3)\)   $:$     \(\displaystyle \forall m, n \in \Z:\)    \(\displaystyle \nu_p^\Z \left({m + n}\right) \)   \(\displaystyle \ge \)   \(\displaystyle \min \left\{ {\nu_p^\Z \left({m}\right), \nu_p^\Z \left({n}\right) }\right\} \)             


Axiom $(V1)$

Let $m, n \in \Z$.

Let $m = 0$ or $n = 0$.

Then:

$\nu_p^\Z \left({m}\right) = +\infty$

or:

$\nu_p^\Z \left({n}\right) = +\infty$

Also:

$n m = 0$

and hence:

\(\displaystyle \nu_p^\Z \left({n m}\right)\) \(=\) \(\displaystyle +\infty\)
\(\displaystyle \) \(=\) \(\displaystyle \nu_p^\Z \left({n}\right) + \nu_p^\Z \left({m}\right)\)


Let $n m \ne 0$.

Then by definition of the restricted $p$-adic valuation:

$p^{\nu_p^\Z \left({n}\right)} \mathop \backslash n$
$p^{\nu_p^\Z \left({n}\right) + 1} \nmid n$

Also:

$p^{\nu_p^\Z \left({m}\right)} \mathop \backslash m$
$p^{\nu_p^\Z \left({m}\right) + 1} \nmid m$


Hence:

$p^{\nu_p^\Z \left({n}\right) + \nu_p^\Z \left({m}\right)} \mathop \backslash n m$
$p^{\nu_p^\Z \left({n}\right) + \nu_p^\Z \left({m}\right) + 1} \nmid n m$

So:

$\nu_p^\Z \left({n m}\right) = \nu_p^\Z \left({n}\right) + \nu_p^\Z \left({m}\right)$

$\Box$


Axiom $(V2)$

By definition of the restricted $p$-adic valuation:

$\forall n \in \Z: \nu_p^\Z \left({n}\right) = +\infty \iff n = 0$

$\blacksquare$


Axiom $(V3)$

Let $m, n \in \Z$.

WLOG let:

$p^\alpha \mathop \backslash n$
$p^\beta \mathop \backslash m$

where $\alpha \ge \beta$.


Then $\exists t \in \Z, k \in \Z$ such that:

\(\displaystyle n + m\) \(=\) \(\displaystyle p^\alpha k + p^\beta t\)
\(\displaystyle \) \(=\) \(\displaystyle p^\beta \left({p^{\alpha - \beta} k + t}\right)\)

Thus:

$p^\beta \mathop \backslash \left({m + n}\right)$

Hence by the definition of $\nu_p^\Z$:

$\nu_p^\Z \left({m + n}\right) \ge \min \left\{{\nu_p^\Z \left({m}\right), \nu_p^\Z \left({n}\right)}\right\}$

$\blacksquare$