## Theorem

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

Let $\Z_p$ be the $p$-adic integers.

Let $a \in \Z_p$.

Let $\ldots a_n \ldots a_3 a_2 a_1 a_0$ be the canonical expansion of $a$.

Then:

$a$ is a $p$-adic unit if and only if $a_0 \ne 0$

## Proof

a is a $p$-adic unit if and only if $\norm a_p = 1 = p^0$

By definition of the canonical expansion:

$a$ is the limit of the $p$-adic expansion $\ds \sum_{n \mathop = 0}^\infty a_n p^n$
$\norm a_p = p^0$ if and only if $a_0 \ne 0$

The result follows.

$\blacksquare$