Properties of Norm on Division Ring/Norm of Negative

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Theorem

Let $\struct {R, +, \circ}$ be a division ring with unity $1_R$.

Let $\norm {\,\cdot\,}$ be a norm on $R$.

Let $x \in R$.


Then:

$\norm {-x} = \norm x$


Proof

By Norm of Negative of Unity:

$\norm {-1_R} = 1$


Then:

\(\ds \norm {-x}\) \(=\) \(\ds \norm {-1_R \circ x}\) Product with Ring Negative
\(\ds \) \(=\) \(\ds \norm {-1_R} \norm x\) Norm Axiom $\text N 2$: Multiplicativity
\(\ds \) \(=\) \(\ds \norm x\) Norm of Negative of Unity

as desired.

$\blacksquare$


Sources