Properties of Norm on Division Ring/Norm of Power Equals Unity
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Theorem
Let $\struct {R, +, \circ}$ be a division ring with zero $0_R$ and unity $1_R$.
Let $\norm {\,\cdot\,}$ be a norm on $R$.
Let $x \in R$.
Then:
- $\forall n \in \N_{>0}: \norm {x^n} = 1 \implies \norm x = 1$
Proof
Let $n \in \N_{>0}$.
Let $\norm {x^n} = 1$.
By Norm Axiom $\text N 2$: Multiplicativity:
- $\norm x^n = 1$
Since $\norm x \ge 0$, by Positive Real Complex Root of Unity:
- $\norm x = 1$
as desired.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.2$: Basic Properties, Lemma 2.2.1 ii)