Valuation Ideal of P-adic Numbers

Theorem
Let $p$ be a prime number.

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

Then the valuation ideal induced by norm $\norm {\,\cdot\,}_p$ is the principal ideal:
 * $p \Z_p = \set {x \in \Q_p: \norm x_p < 1}$

where $\Z_p$ denotes the $p$-adic integers.

Proof
From P-adic Integers is Local Ring, $\Z_p$ is a local ring.

From Principal Ideal from Element in Center of Ring, $p \Z_p$ is a principal ideal.

Now:

Hence:
 * $p \Z_p = \set {x \in \Q_p: \norm x_p < 1}$