P-adic Number times P-adic Norm is P-adic Unit
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Theorem
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p^\times$ be the $p$-adic units.
Let $a \in \Q_p$.
Then there exists $n \in \Z$ such that:
- $p^n a \in \Z_p^\times$
where
- $p^n = \norm a_p$
Proof
From P-adic Norm of p-adic Number is Power of p, there exists $v \in \Z$ such that $\norm a_p = p^{-v}$.
Let $n = -v$.
Now:
\(\ds \norm{p^n a}_p\) | \(=\) | \(\ds \norm{p^n }_p \norm a_p\) | Norm axiom (N2) (Multiplicativity) | |||||||||||
\(\ds \) | \(=\) | \(\ds p^{-n} \norm a_p\) | Definition of $p$-adic norm on the rational numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds p^{-n} p^{n}\) | Definition of $n$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds p^n a\) | \(\in\) | \(\ds \Z_p^\times\) | P-adic Unit has Norm Equal to One |
$\blacksquare$
Sources
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.5$ Arithmetical operations in $\Q_p$: Proposition $1.37$