# Field Norm on 5th Cyclotomic Ring

## Theorem

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

Let $\alpha = a + i b \sqrt 5$ be an arbitrary element of $\Z \sqbrk {i \sqrt 5}$.

The field norm of $\alpha$ is given by:

$\map N \alpha = a^2 + 5 b^2$

## Proof

 $\displaystyle \map N \alpha$ $=$ $\displaystyle \cmod \alpha^2$ Definition of Field Norm of Complex Number $\displaystyle$ $=$ $\displaystyle \paren {\sqrt {a^2 + \paren {b \sqrt 5}^2} }^2$ Definition of Complex Modulus $\displaystyle$ $=$ $\displaystyle a^2 + 5 b^2$

$\blacksquare$