Field Norm on 5th Cyclotomic Ring
Jump to navigation
Jump to search
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
\(\ds \map N \alpha\) | \(=\) | \(\ds \cmod \alpha^2\) | Definition of Field Norm of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sqrt {a^2 + \paren {b \sqrt 5}^2} }^2\) | Definition of Complex Modulus | |||||||||||
\(\ds \) | \(=\) | \(\ds a^2 + 5 b^2\) |
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $19$