Properties of Norm on Division Ring/Norm of Power Equals Unity

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:
 * $\forall n \in \N_{\gt 0}: \norm {x^n} = 1 \implies \norm x = 1$

Proof
Let $n \in \N_{\gt 0}$.

Let $\norm {x^n} = 1$.

By Norm axiom (N2) (Multiplicativity) then:
 * $\norm x^n = 1$

Since $\norm x \ge 0$, by Positive Real Complex Root of Unity then:
 * $\norm x = 1$

as desired.