Value of Field Norm on 5th Cyclotomic Ring is Integer

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 $\Z \sqbrk {i \sqrt 5}$.

Let $\map N \alpha$ denoted the field norm of $\alpha$.

Then $\map N \alpha$ is an integer.

Proof
From Field Norm on 5th Cyclotomic Ring:
 * $\map N \alpha = a^2 + 5 b^2$

From the definition of the $5$th cyclotomic ring:
 * $\Z \sqbrk {i \sqrt 5} = \set {a + i \sqrt 5 b: a, b \in \Z}$

That is, both $a$ and $b$ are integers.

Hence $a^2 + 5 b^2$ is also an integer.