---
tag: TWG8QKY8TA
kind: definition
name: Cyclotomic extension
msc: [12F10, 11R18]
defines: VD6VC3ZMMY
uses: [0WQ1A8SHTZ, 0J817RTVG6, 0S61N75H45, 02NZZ1J0FH]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclotomic_field
---

## Definition

Let $K$ be a field and let $n \ge 1$ be an integer that is not divisible by the characteristic of $K$.
A [[field extension|0WQ1A8SHTZ]] $L/K$ is an *$n$-th cyclotomic extension* if both conditions hold:

1. $L$ contains a [[primitive $n$-th root of unity|0J817RTVG6]] $\zeta_n$;
2. $L = K(\zeta_n)$.

Equivalently, $L$ is a [[splitting field|02NZZ1J0FH]] of $x^n - 1$ over $K$.
More generally, let $S$ be a set of positive integers.
Then $L/K$ is an *$S$-cyclotomic extension* if $L$ contains a primitive $n$-th root of unity for each $n \in S$.
Moreover $L$ must be generated over $K$ by the [[roots of unity|0S61N75H45]] of these orders.

## Remarks

Example: $\Q(i)/\Q$ is a $4$-th cyclotomic extension, since $i$ has order $4$ in $\C^\times$ and $\Q(i)$ is generated by $i$.
Non-example: $\Q(\sqrt[3]{2})/\Q$ is not a $3$-rd cyclotomic extension, since $\Q(\sqrt[3]{2})$ is a subfield of $\R$ and contains no primitive cube root of unity.
The two forms agree because the roots of $x^n - 1$ are the powers of $\zeta_n$.
An $n$-th cyclotomic extension is Galois, and $\sigma \mapsto k$ with $\sigma(\zeta_n) = \zeta_n^k$ is an injective homomorphism $\Gal(L/K) \to (\Z/n\Z)^\times$.
For $K = \Q$ this map is an isomorphism, and $L$ is the cyclotomic field $\Q(\zeta_n)$.