P-adic Unit has Norm Equal to One

Theorem

Let $p$ be a prime number.

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

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

Let $x \in \Q_p$.

Then x is a $p$-adic unit if and only if $\norm x_p = 1$

Proof

Necessary Condition

Let $x$ be a $p$-adic unit.

Then:

$x \in \Z_p$

$x^{-1} \in \Z_p$

By definition of the $p$-adic integers:

$\norm x_p \le 1$

$\norm {x^{-1} }_p \le 1$

From Norm of Inverse in Division Ring:

$\norm x_p \ge 1$

It follows that:

$\norm x_p = 1$

$\Box$

Sufficient Condition

Let $\norm x_p = 1$.

From Norm of Inverse in Division Ring:

$\norm {x^{-1} }_p = 1$

By definition of the $p$-adic integers:

$x \in \Z_p$

$x^{-1} \in \Z_p$

It follows that $x$ is an invertible element in $\Z_p$.

That is, x is a $p$-adic unit.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.5$ Arithmetical operations in $\Q_p$: Exercise $26$