Valuation Ideal is Maximal Ideal of Induced Valuation Ring/Corollary 1

Theorem
Let $\struct {R, \norm{\,\cdot\,}}$ be a non-Archimedean normed division ring with zero $0_R$ and unity $1_R$.

Let $\mathcal O$ be the valuation ring induced by the non-Archimedean norm $\norm{\,\cdot\,}$, that is:
 * $\mathcal O = \set{x \in R : \norm{x} \le 1}$

Then:
 * $\mathcal O$ is a local ring.