\begin{definition}[Definition:Field Norm of Quaternion]
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 '''norm''' of $\mathbf x$ is the real number defined as:
:$n \left({\mathbf x}\right) := \left\vert{\mathbf x \overline {\mathbf x} }\right\vert = \left\vert{\overline {\mathbf x} \mathbf x }\right\vert = a^2 + b^2 + c^2 + d^2$
\end{definition}