P-adic Valuation Extends to P-adic Numbers

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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 valuation that extends $\nu_p^\Q$ from $\Q$ to $\Q_p$.


Proof

It needs to be shown that $\nu_p$:

$(1): \quad \nu_p$ is a mapping into $\Z \cup \set {+\infty}$
$(2): \quad \nu_p$ satisfies the valuation axioms $\text V 1$, $\text V 2$ and $\text V 3$
$(3): \quad \nu_p$ extends $\nu_p^\Q$.


Let $x, y \in \Q_p$.


$\nu_p$ is a mapping into $\Z \cup \set {+\infty}$

If $x = 0$ then $\map {\nu_p} x = +\infty$ by definition.


Let $x \ne 0$.

By P-adic Norm of p-adic Number is Power of p then:

$\exists v \in \Z: \norm x_p = p^{-v}$

Hence:

\(\ds \map {\nu_p} x\) \(=\) \(\ds -\log_p \norm x_p\) Since $x \ne 0$
\(\ds \) \(=\) \(\ds -\log_p p^{-v}\) Definition of $v$
\(\ds \) \(=\) \(\ds -\paren {-v}\) Definition of Real General Logarithm
\(\ds \) \(=\) \(\ds v\)
\(\ds \) \(\in\) \(\ds \Z\) Definition of $v$

$\Box$


$\nu_p$ satisfies $(\text V 1)$

If $x = 0$ then:

\(\ds \map {\nu_p} {0 \cdot y}\) \(=\) \(\ds \map {\nu_p} 0\)
\(\ds \) \(=\) \(\ds +\infty\) Definition of $\nu_p$
\(\ds \) \(=\) \(\ds +\infty \cdot \map {\nu_p} y\) Definition of Extended Real Multiplication
\(\ds \) \(=\) \(\ds \map {\nu_p} 0 \cdot \map {\nu_p} y\) Definition of $\nu_p$


Similarly, if $y = 0$ then:

\(\ds \map {\nu_p} {x \cdot 0}\) \(=\) \(\ds \map {\nu_p} x \cdot \map {\nu_p} 0\)


If $x \ne 0, y \ne 0$ then:

\(\ds \map {\nu_p} {x y}\) \(=\) \(\ds -\log \norm {x y}_p\) $x y \ne 0$
\(\ds \) \(=\) \(\ds -\log \norm x_p \norm y_p\) Non-Archimedean Norm Axiom $\text N 2$: Multiplicativity
\(\ds \) \(=\) \(\ds -\paren {\log \norm x_p + \log \norm y_p}\) Sum of General Logarithms
\(\ds \) \(=\) \(\ds \paren {-\log \norm x_p} + \paren {-\log \norm y_p}\)
\(\ds \) \(=\) \(\ds \map {\nu_p} x + \map {\nu_p} y\) Definition of $\nu_p$

$\Box$


$\nu_p$ satisfies $(\text V 2)$

If $x = 0$ then $\map {\nu_p} x = +\infty$ by definition.

If $x \ne 0$ then $\map {\nu_p} x \in \Z$ by the above.

Hence:

$\map {\nu_p} x = +\infty \iff x = 0$

$\Box$


$\nu_p$ satisfies $(\text V 3)$

Suppose $x = 0$.

Then:

\(\ds \map {\nu_p} {0 + y}\) \(=\) \(\ds \map {\nu_p} y\)
\(\ds \) \(\ge\) \(\ds \min \set {\map {\nu_p} 0, \map {\nu_p} y}\) Definition of Min Operation


Similarly, if $y = 0$ then:

\(\ds \map {\nu_p} {x + 0}\) \(\ge\) \(\ds \min \set {\map {\nu_p} x, \map {\nu_p} 0}\)


Suppose $x + y = 0$.

Then:

\(\ds \map {\nu_p} {x + y}\) \(=\) \(\ds \map {\nu_p} 0\)
\(\ds \) \(=\) \(\ds +\infty\) Definition of $\nu_p$
\(\ds \) \(\ge\) \(\ds \map {\nu_p} x\) Definition of Extended Real Number Line
\(\ds \) \(\ge\) \(\ds \min \set {\map {\nu_p} x, \map {\nu_p} y}\) Definition of Min Operation


Suppose $x \ne 0, y \ne 0, x + y \ne 0$.

Then:

\(\ds \norm {x + y}\) \(\le\) \(\ds \max \set {\norm x_p, \norm y_p}\) Non-Archimedean Norm Axiom $\text N 4$: Ultrametric Inequality
\(\ds \) \(\) \(\ds \)
\(\ds \leadsto \ \ \) \(\ds \log \norm {x + y}\) \(\le\) \(\ds \log \max \set {\norm x_p, \norm y_p}\) Logarithm is Strictly Increasing
\(\ds \) \(=\) \(\ds \max \set {\log \norm x_p, \log \norm y_p}\) Logarithm is Strictly Increasing
\(\ds \) \(\) \(\ds \)
\(\ds \leadsto \ \ \) \(\ds -\log \norm {x + y}\) \(\ge\) \(\ds -\max \set {\log \norm x_p, \log \norm y_p}\) Inversion Mapping Reverses Ordering in Ordered Group
\(\ds \) \(=\) \(\ds \min \set {-\log \norm x_p, -\log \norm y_p}\)
\(\ds \) \(\) \(\ds \)
\(\ds \leadsto \ \ \) \(\ds \map {\nu_p} {x + y}\) \(\ge\) \(\ds \min \set {\map {\nu_p} x, \map {\nu_p} y}\) Definition of $\nu_p$

$\Box$


$\nu_p$ extends $\nu_p^\Q$

Let $x \in \Q$.

If $x = 0$ then $\map {\nu_p} 0 = +\infty = \map {\nu_p^\Q} 0$.


Let $x \ne 0$.

From Rational Numbers are Dense Subfield of P-adic Numbers:

the $p$-adic norm $\norm {\,\cdot\,}_p$ on $p$-adic numbers $\Q_p$ is an extension of the $p$-adic norm $\norm {\,\cdot\,}_p$ on rational numbers $\Q$ by definition.

Hence:

\(\ds \map {\nu_p} x\) \(=\) \(\ds -\log \norm x_p\) Definition of $\nu_p$
\(\ds \) \(=\) \(\ds \map {\nu_p^\Q} x\) Definition of $p$-adic norm $\norm {\,\cdot\,}_p$ on rational numbers $\Q$

$\blacksquare$


Sources

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