Characterization of Rational P-adic Unit

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Theorem

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

Let $\Z^\times_p$ be the $p$-adic units.

Let $\Q$ be the rational numbers.


Then:

$\Z^\times_p \cap \Q = \set{\dfrac a b \in \Q : p \nmid ab}$

Proof

Let $\norm{\,\cdot\,}^\Q _p$ denote the $p$-adic norm on the rational numbers.


We have:

\(\ds \Z^\times_p \cap \Q\) \(=\) \(\ds \set{\dfrac a b \in \Q : \norm {\dfrac a b}_p = 1}\) P-adic Unit has Norm Equal to One
\(\ds \) \(=\) \(\ds \set{\dfrac a b \in \Q : \norm {\dfrac a b}_p \le 1} \setminus \set{\dfrac a b \in \Q : \norm {\dfrac a b}_p < 1}\) Definition of Set Difference
\(\ds \) \(=\) \(\ds \set{\dfrac a b \in \Q : \norm{\dfrac a b}^\Q_p \le 1} \setminus \set{\dfrac a b \in \Q : \norm {\dfrac a b}^\Q_p < 1}\) Rational Numbers are Dense Subfield of P-adic Numbers
\(\ds \) \(=\) \(\ds \set{\dfrac a b \in \Q : \norm{\dfrac a b}^\Q_p \le 1} \setminus \set{\dfrac a b \in \Q : p \nmid b, p \divides a}\) Valuation Ideal of P-adic Norm on Rationals
\(\ds \) \(=\) \(\ds \set{\dfrac a b \in \Q : p \nmid b} \setminus \set{\dfrac a b \in \Q : p \nmid b, p \divides a}\) Valuation Ring of P-adic Norm on Rationals
\(\ds \) \(=\) \(\ds \set{\dfrac a b \in \Q : p \nmid b, p \nmid a}\) Definition of Set Difference
\(\ds \) \(=\) \(\ds \set{\dfrac a b \in \Q : p \nmid ab}\) Prime Divisor of Coprime Integers and Divisor Divides Multiple

$\blacksquare$

Sources

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