---
tag: BHKM3A1GB6
kind: definition
name: Cyclotomic field
msc: [11R18]
defines: NN77VYSGXR
uses: [MWBDFS2KJ5, 0EHM5ZP5QH, 08KEYFAH66, 0DAMHCT17R, 0SK1VHZ5R2]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclotomic_field
---

## Definition

Let $n \ge 1$ be an integer and let $\zeta_n = e^{2\pi i/n} \in \C$, a [[primitive $n$-th root of unity|0EHM5ZP5QH]].
The *$n$-th cyclotomic field* is the subfield $\Q(\zeta_n)$ of $\C$ generated by $\zeta_n$ over $\Q$.
A *cyclotomic field* is a field of this form.

## Remarks

The minimal polynomial of $\zeta_n$ over $\Q$ is the [[cyclotomic polynomial|08KEYFAH66]] $\Phi_n$, which is irreducible over $\Q$ by the [[irreducibility of cyclotomic polynomials|0SK1VHZ5R2]].
So $[\Q(\zeta_n) : \Q] = \varphi(n)$, where $\varphi$ is [[Euler's totient function|0DAMHCT17R]], and $\Q(\zeta_n)$ is an [[algebraic number field|MWBDFS2KJ5]].
Example: for $n = 4$, $\zeta_4 = i$ and $\Phi_4 = x^2 + 1$, so $\Q(\zeta_4) = \Q(i)$ has degree $\varphi(4) = 2$.
Non-example: $\Q(\sqrt[3]{2})$ is not a cyclotomic field: every cyclotomic field is a Galois extension of $\Q$, but $\Q(\sqrt[3]{2}) \subset \R$ does not contain the non-real roots of $x^3 - 2$.
The Galois group of $\Q(\zeta_n)/\Q$ is isomorphic to $(\Z/n\Z)^{\times}$, with $a$ acting by $\zeta_n \mapsto \zeta_n^a$.
The ring of integers of $\Q(\zeta_n)$ is $\Z[\zeta_n]$.
For odd $n$, $\Q(\zeta_{2n}) = \Q(\zeta_n)$, because $-\zeta_n$ is a primitive $2n$-th root of unity.