Field Norm on 5th Cyclotomic Ring

Theorem
Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring.

Let $\alpha = a + i b \sqrt 5$ be an arbitrary element of $\Z \sqbrk {i \sqrt 5}$.

The field norm of $\alpha$ is given by:
 * $\map N \alpha = a^2 + 5 b^2$