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

$\blacksquare$