Valuation Ring of P-adic Norm on Rationals/Corollary 1

Theorem

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.

Let $\OO$ be the induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$.

The set of integers $\Z$ is a subring of $\OO$.

Proof

By Valuation Ring of P-adic Norm on Rationals, the induced valuation ring $\OO$ is the set:

$\OO = \Z_{\paren p} = \set {\dfrac a b \in \Q : p \nmid b}$

Since $p \nmid 1$ then for all $a \in \Z$, $a = \dfrac a 1 \in \OO$.

Hence $\Z \subseteq \OO$.

By Valuation Ring of Non-Archimedean Division Ring is Subring then $\OO$ is a subring of $\Q$.

By Integers form Subdomain of Rationals then $\Z$ is a subring of $\Q$.

By Intersection of Subrings is Largest Subring Contained in all Subrings then $\Z \cap \OO = \Z$ is a subring of $\OO$.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 2.4$ Algebra, Problem $59$