Field Norm of Quaternion is Multiplicative

Theorem
Let $\mathbf x$ 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 a multiplicative function.