Elements of 5th Cyclotomic Ring with Field Norm 1

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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.

$\blacksquare$


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