P-adic Valuation Extends to P-adic Numbers

Theorem
Let $p$ be a prime number.

Let $\nu_p^\Q: \Q \to \Z \cup \set {+\infty}$ be the $p$-adic valuation on the set of rational numbers.

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

Let $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ be defined by:
 * $\forall x \in \Q_p : \map {\nu_p} x = \begin{cases}

\log_p \norm {x}_p : x \ne 0 \\ +\infty : x = 0 \end{cases}$

Then $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ is a unique valuation that extends $\nu_p^\Q$ to $\nu_p$.