Units of 5th Cyclotomic Ring

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

The units of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ are $1$ and $-1$.

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

Let $z_1 \in \Z \sqbrk {i \sqrt 5}$ be a unit of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$.

Thus by definition:
 * $\exists z_2 \in \Z \sqbrk {i \sqrt 5}: z_1 \times z_2 = 1$

Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$.

Then:

The result follows.