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 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
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$: Lemma $3.3.2$