Image of P-adic Norm

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Theorem

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$.

Then the image of $\norm {\,\cdot\,}_p$ is:

$\Img {\norm {\,\cdot\,}_p} = \set {p^n : n \in \Z} \cup \set 0$

Proof

This follows immediately from:

Definition of P-adic Norm

Definition of P-adic Valuation

$\blacksquare$