# Characterization of Rational P-adic Unit

## 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

$\blacksquare$