Valuation Ring of P-adic Norm on Rationals/Corollary 1
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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 $59$