Definition:Cyclotomic Ring
Definition
Let $\Z \sqbrk {i \sqrt n}$ be the set $\set {a + i b \sqrt n: a, b \in \Z}$.
The algebraic structure $\struct {\Z \sqbrk {i \sqrt n}, +, \times}$ is the $n$th cyclotomic ring.
Seriously unsure about this. Picked the name by back formation from Definition:Cyclotomic Field and found a tiny number of papers on the internet referring to such a construct by that name. "Ring of integers on cyclotomic field" is another way of describing it. The book I'm working through provides the example $\Z \sqbrk {i \sqrt 5}$ in one of its final few exercises so its coverage it sketchy. Does anyone know about this?
If you are fairly certain that all issues are dealt with, please remove {{proofread}} from the code.
Examples
$5$th Cyclotomic Ring
The $5$th cyclotomic ring is the algebraic structure:
- $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$
where $\Z \sqbrk {i \sqrt 5}$ is the set $\set {a + i b \sqrt 5: a, b \in \Z}$.
$\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ is a ring.
Also see
- Results about cyclotomic rings can be found here.