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

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Theorem

Let $p$ be a prime number.

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

Let $\Z_p$ 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:

$p^0 a \in \Z_p$.


Case: $\norm{a}_p > 1$

Let $\norm{a}_p > 1$.

From P-adic Norm of p-adic Number is Power of p, there exists $v \in \Z$ such that $\norm a_p = p^{-v}$.

Hence:

$p^{-v} > 1 = p^0$.

From Real Power Function on Base Greater than One is Strictly Increasing:

$-v > 0$.


Let $n = -v$.

Then:

\(\displaystyle \norm{p^n a}_p\) \(=\) \(\displaystyle \norm{p^n }_p \norm a_p\) Norm axiom (N2) (Multiplicativity)
\(\displaystyle \) \(=\) \(\displaystyle p^{-n} \norm a_p\) Definition of $p$-adic norm on the rational numbers
\(\displaystyle \) \(=\) \(\displaystyle p^{-n} p^{n}\) By definition of $n$
\(\displaystyle \) \(=\) \(\displaystyle 1\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle p^n a\) \(\in\) \(\displaystyle \Z_p\) Definition of $p$-adic integers


The result follows.

$\blacksquare$

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