Valuation Ring of Non-Archimedean Division Ring is Clopen

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Theorem

Let $\struct {R, \norm {\,\cdot\,} }$ be a non-Archimedean normed division ring with zero $0_R$.

Let $\OO$ be valuation ring induced by $\norm{\,\cdot\,}$.


Then $\OO$ is a both open and closed in the metric induced by $\norm{\,\cdot\,}$.


Corollary

Let $p$ be a prime number.

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


Then the $p$-adic integers $\Z_p$ is both open and closed in the $p$-adic metric.


Proof

The valuation ring $\OO$ Is the open ball $\map {B_1^-} {0_R}$ by definition.

By Open Balls of Non-Archimedean Division Rings are Clopen then $\OO$ is both open and closed in the metric induced by $\norm {\,\cdot\,}$.

$\blacksquare$