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