P-adic Norm forms Non-Archimedean Valued Field/Rational Numbers
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Theorem
The $p$-adic norm $\norm{\,\cdot\,}_p$ forms a non-Archimedean norm on the rational numbers $\Q$.
The rational numbers $\struct{\Q, \norm{\,\cdot\,}_p}$ with the $p$-adic norm is a valued field with a non-Archimedean norm.
Proof
First we note that the $p$-adic norm is a norm.
Let $\nu_p$ denote the $p$-adic valuation on the rational numbers.
Recall the definition of the $p$-adic norm:
- $\forall q \in \Q: \norm q_p := \begin{cases} 0 & : q = 0 \\ p^{-\map {\nu_p} q} & : q \ne 0 \end{cases}$
We must show the following holds for all $x, y \in \Q$:
- $\norm {x + y}_p \le \max \set {\norm x_p, \norm y_p}$
If $x = 0$ or $y = 0$, or $x + y = 0$, the result is trivial, as follows:
Let $x = 0$.
Then:
\(\ds x\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \norm x_p\) | \(=\) | \(\ds 0\) | Definition of $p$-adic Norm | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \max \set {\norm x_p, \norm y_p}\) | \(=\) | \(\ds \norm y_p\) | as $\norm y_p \ge 0 = \norm x_p$ from Non-Archimedean Norm Axiom $\text N 1$: Positive Definiteness | ||||||||||
\(\ds \) | \(=\) | \(\ds \norm {x + y}_p\) |
and so $\norm {x + y}_p \le \max \left( \norm x_p, \norm y_p \right)$
The same argument holds for $y = 0$.
Let $x + y = 0$.
\(\ds x + y\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \norm {x + y}_p\) | \(=\) | \(\ds 0\) | Definition of $p$-adic Norm | ||||||||||
\(\ds \) | \(\le\) | \(\ds \max \set {\norm x_p, \norm y_p}\) | as $\norm x_p \ge 0$ and $\norm y_p \ge 0$ from Non-Archimedean Norm Axiom $\text N 1$: Positive Definiteness |
Let $x, y, x + y \in \Q_{\ne 0}$.
From $p$-adic Valuation is Valuation:
- $\map {\nu_p} {x + y} \ge \min \set {\map {\nu_p} x, \map {\nu_p} y}$
Then:
\(\ds \norm {x + y}_p\) | \(=\) | \(\ds p^{-\map {\nu_p} {x + y} }\) | Definition of $p$-adic Norm | |||||||||||
\(\ds \) | \(\le\) | \(\ds \max \set {p^{- \map {\nu_p} x}, p^{-\map {\nu_p} y} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \max \set {\norm x_p, \norm y_p}\) | Definition of $p$-adic Norm |
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.1$: Absolute Values on a Field: Proposition $2.1.5$
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.4$ The field of $p$-adic numbers $\Q_p$: Proposition $1.26$