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.

Proof

\(\ds \map n {\mathbf x \mathbf y}\)

\(=\)

\(\ds \mathbf x \, \mathbf y \ \overline {\mathbf x \, \mathbf y}\)

Definition of Field Norm of Quaternion

\(\ds \)

\(=\)

\(\ds \mathbf x \, \mathbf y \paren {\overline {\mathbf y} \, \overline {\mathbf x} }\)

Product of Quaternion Conjugates

\(\ds \)

\(=\)

\(\ds \mathbf x \paren {\mathbf y \paren {\overline {\mathbf y} \, \overline {\mathbf x} } }\)

\(\ds \)

\(=\)

\(\ds \mathbf x \paren {\paren {\mathbf y \, \overline {\mathbf y} } \overline {\mathbf x} }\)

\(\ds \)

\(=\)

\(\ds \mathbf x \paren {\map n {\mathbf y} \overline {\mathbf x} }\)

\(\ds \)

\(=\)

\(\ds \map n {\mathbf y} \paren {\mathbf x \, \overline {\mathbf x} }\)

\(\ds \)

\(=\)

\(\ds \map n {\mathbf y} \, \map n {\mathbf x}\)

\(\ds \)

\(=\)

\(\ds \map n {\mathbf x} \, \map n {\mathbf y}\)

$\blacksquare$

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem