5th Cyclotomic Ring has no Elements with Field Norm of 2 or 3

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

There are no elements of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ whose field norm is either $2$ or $3$.

Proof
Let $\map N z$ denote the field norm of $z \in \Z \sqbrk {i \sqrt 5}$.

Let $z = x + i y$.

Then:

But Square Root of Prime is Irrational so $x \notin \Z$.

Similarly for $\map N z = 3$.