## Theorem

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\Z_p^\times$ be the $p$-adic units.

Let $a \in \Q_p$.

Then there exists $n \in \Z$ such that:

$p^n a \in \Z_p^\times$

where

$p^n = \norm a_p$

## Proof

From P-adic Norm of p-adic Number is Power of p, there exists $v \in \Z$ such that $\norm a_p = p^{-v}$.

Let $n = -v$.

Now:

 $\ds \norm{p^n a}_p$ $=$ $\ds \norm{p^n }_p \norm a_p$ Norm axiom (N2) (Multiplicativity) $\ds$ $=$ $\ds p^{-n} \norm a_p$ Definition of $p$-adic norm on the rational numbers $\ds$ $=$ $\ds p^{-n} p^{n}$ Definition of $n$ $\ds$ $=$ $\ds 1$ $\ds \leadsto \ \$ $\ds p^n a$ $\in$ $\ds \Z_p^\times$ P-adic Unit has Norm Equal to One

$\blacksquare$