P-adic Number times Integer Power of p is P-adic Integer

Theorem

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\Z_p$ be the $p$-adic integers.

Then:

$\forall a \in \Q_p: \exists n \in \N: p^n a \in \Z_p$

Proof

Let $a \in \Q_p$.

Case: $\norm a_p \le 1$

Let $\norm a_p \le 1$.

By definition of the $p$-adic integers:

$a \in \Z_p$.

Hence:

$p^0 a \in \Z_p$.

$\Box$

Case: $\norm a_p > 1$

Let $\norm a_p > 1$.

From P-adic Number times P-adic Norm is P-adic Unit, there exists $n \in \Z$ such that $p^n a$ is a $p$-adic unit.

Then:

\(\ds \norm{p^n a}_p\)

\(=\)

\(\ds 1\)

P-adic Unit has Norm Equal to One

\(\, \ds \leadstoandfrom \, \)

\(\ds \norm{p^n}_p \norm a_p\)

\(=\)

\(\ds 1\)

Non-Archimedean Norm Axiom $\text N 2$: Multiplicativity

\(\, \ds \leadstoandfrom \, \)

\(\ds \norm{p^n}_p\)

\(<\)

\(\ds 1\)

As $\norm{a}_p > 1$ by assumption

\(\, \ds \leadstoandfrom \, \)

\(\ds p^{-n}\)

\(<\)

\(\ds 1\)

Definition of $p$-adic norm on integers

\(\, \ds \leadstoandfrom \, \)

\(\ds p^n\)

\(>\)

\(\ds 1\)

\(\, \ds \leadstoandfrom \, \)

\(\ds n\)

\(>\)

\(\ds 0\)

Real Power Function on Base Greater than One is Strictly Increasing

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$: Lemma $3.3.5$