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 $\norm x_p = 1$

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:
 * $\norm x_p = 1$

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.