Value of Field Norm on 5th Cyclotomic Ring is Integer
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 $\Z \sqbrk {i \sqrt 5}$.
Let $\map N \alpha$ denoted the field norm of $\alpha$.
Then $\map N \alpha$ is an integer.
Proof
From Field Norm on 5th Cyclotomic Ring:
- $\map N \alpha = a^2 + 5 b^2$
From the definition of the $5$th cyclotomic ring:
- $\Z \sqbrk {i \sqrt 5} = \set {a + i \sqrt 5 b: a, b \in \Z}$
That is, both $a$ and $b$ are integers.
Hence $a^2 + 5 b^2$ is also an integer.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $19$