# Elements of 5th Cyclotomic Ring with Field Norm 1

## Theorem

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

The only elements of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ whose field norm equals $1$ are the units of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$: $1$ and $-1$.

## Proof

From Units of 5th Cyclotomic Ring, $1$ and $-1$ are the only units of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$.

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

Let $z \in \Z \sqbrk {i \sqrt 5}$ such that $\map N z = 1$.

Let $z = x + i y$.

Then:

 $\displaystyle \map N z$ $=$ $\displaystyle 1$ Field Norm on 5th Cyclotomic Ring $\displaystyle \leadsto \ \$ $\displaystyle x^2 + 5 y^2$ $=$ $\displaystyle 1$ $\displaystyle \leadsto \ \$ $\displaystyle x^2$ $=$ $\displaystyle 1$ $\, \displaystyle \land \,$ $\displaystyle y$ $=$ $\displaystyle 0$

The result follows.

$\blacksquare$