Field Norm of Quaternion is not Norm

Theorem

Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.

Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$.

The field norm of $\mathbf x$:

$\map n {\mathbf x} := \size {\mathbf x \overline {\mathbf x} }$

is not a norm in the abstract algebraic context of a division ring.

Proof

Each of the norm axioms is examined in turn:

Norm Axiom $\text N 1$: Positive Definiteness

This is proved in Field Norm of Quaternion is Positive Definite‎.

$\Box$

Norm Axiom $\text N 2$: Multiplicativity

This is proved in Field Norm of Quaternion is Multiplicative.

$\Box$

Norm Axiom $\text N 3$: Triangle Inequality

For example:

$\map n {1 + 1} = 4 > 2 = \map n 1 + \map n 1$

and so Norm Axiom $\text N 3$: Triangle Inequality is not satisfied.

$\Box$

Not all the norm axioms are fulfilled.

Hence the result.

$\blacksquare$