Subring of Non-Archimedean Division Ring

From ProofWiki
Jump to navigation Jump to search

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 $\norm {\, \cdot \,}_S$

$\blacksquare$