# Definition:Cyclotomic Ring

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## 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**.

This page needs proofreading.In particular: 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 believe all issues are dealt with, please remove `{{Proofread}}` from the code.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{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**.