Properties of Norm on Division Ring/Norm of Negative of Unity

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
By Product of Ring Negatives then:
 * $-1_R \circ -1_R = 1_R \circ 1_R=1_R$.

So:

Thus:


 * $\norm{-1_R} = \pm 1$

By the norm axiom (N1) (Positive defintiteness) then:
 * $\norm{-1_R} \ge 0$

Hence:
 * $\norm{-1_R} = 1$