Field Norm on 5th Cyclotomic Ring

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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
\(\displaystyle \) \(=\) \(\displaystyle a^2 + 5 b^2\)

$\blacksquare$


Sources