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 $-1$.

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:

$\ds 1$

$=$

$\ds \map N 1$

$\ds $

$=$

$\ds \map N {z_1 \times z_2}$

$\ds $

$=$

$\ds \map N {z_1} \times \map N {z_2}$

$\ds $

$=$

$\ds \paren { {x_1}^2 + 5 {y_1}^2} \paren { {x_2}^2 + 5 {y_2}^2}$

$\ds \leadsto \ \ $

$\ds {x_1}^2$

$=$

$\ds 1$

$\, \ds \land \, $

$\ds {x_2}^2$

$=$

$\ds 1$

$\, \ds \land \, $

$\ds y_1$

$=$

$\ds y_2 = 0$

The result follows.

$\blacksquare$