Equivalence of Definitions of P-adic Valuation on P-adic Numbers

Theorem

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

The following definitions of the concept of P-adic Valuation on P-adic Numbers are equivalent:

Definition 1

The $p$-adic valuation on $p$-adic numbers is the function $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ 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}$

Definition 2

The $p$-adic valuation on $p$-adic numbers is the mapping $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ defined by:

$(1): \quad \map {\nu_p} 0 = +\infty$

$(2): \quad $for all $x \in \Q_p \setminus \set 0$:

$\map {\nu_p} x$ is the index of the first non-zero coefficient in the canonical $p$-adic expansion of $x$

Proof

Let $x \in \Q_p \setminus \set 0$.

Let $l$ be the index of the first non-zero coefficient in the $p$-adic expansion:

$l = \min \set {i: i \ge m \land d_i \ne 0}$

From P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient:

$\norm x_p = p^{-l}$

By definition of real general logarithm:

$-\log_p \norm x_p = l$

The result follows.

$\blacksquare$