P-adic Expansion of P-adic Unit

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

From P-adic Unit has Norm Equal to One:

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 $\ds \sum_{n \mathop = 0}^\infty a_n p^n$

From P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient:

$\norm a_p = p^0$ if and only if $a_0 \ne 0$

The result follows.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$: Problem $105$
• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.5$ Arithmetical operations in $\Q_p$: Proposition $1.36$