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.

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

Proof
By Principal Ideal from Element in Center of Ring then $p\Z_p$ is a principal ideal.

Now:

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