Subring of Non-Archimedean Division Ring
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Theorem
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with non-archimedean norm $\norm {\, \cdot \,}$.
Let $\struct {S, \norm {\, \cdot \,}_S }$ be a normed division subring of $R$.
Then:
- $\norm {\, \cdot \,}_S$ is a non-archimedean norm.
Proof
$\forall x, y \in S$:
\(\ds \norm {x + y}_S\) | \(=\) | \(\ds \norm {x + y}\) | Definition of $\norm {\,\cdot\,}_S$ | |||||||||||
\(\ds \) | \(\le\) | \(\ds \max \set {\norm x, \norm y}\) | $(\text N 4)$: Ultrametric Inequality | |||||||||||
\(\ds \) | \(=\) | \(\ds \max \set {\norm x_S, \norm y_S}\) | Definition of $\norm {\, \cdot \,}_S$ |
$\blacksquare$