Properties of Norm on Division Ring/Norm of Negative of Unity
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Theorem
Let $\struct {R, +, \circ}$ be a division ring with unity $1_R$.
Let $\norm {\,\cdot\,}$ be a norm on $R$.
Then:
- $\norm {-1_R} = 1$
Proof
- $-1_R \circ -1_R = 1_R \circ 1_R = 1_R$
So:
\(\ds \norm {-1_R}^2\) | \(=\) | \(\ds \norm {-1_R} \norm {-1_R}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \norm {-1_R \circ -1_R}\) | Norm Axiom $\text N 2$: Multiplicativity | |||||||||||
\(\ds \) | \(=\) | \(\ds \norm {1_R}\) | Product of Ring Negatives | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | Norm of Unity of Division Ring |
Thus:
- $\norm {-1_R} = \pm 1$
By Norm Axiom $\text N 1$: Positive Definiteness:
- $\norm {-1_R} \ge 0$
Hence:
- $\norm {-1_R} = 1$
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.2$: Basic Properties, Lemma $2.2.1\,iii)$
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$: Normed Fields, Theorem $1.6\,(a)$