---
tag: VD6VC3ZMMY
kind: concept
name: Cyclotomic extension
msc: [12F10, 11R18]
specializes: [EQDEXBWGJH]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclotomic_field
---

## Summary

A cyclotomic extension of a field $K$ is obtained by adjoining [[roots of unity|0S61N75H45]] to $K$.
The extension $K(\zeta_n)/K$ generated by a [[primitive $n$-th root of unity|0J817RTVG6]] is Galois, and its Galois group embeds in $(\Z/n\Z)^\times$, so it is [[abelian|EQDEXBWGJH]].
Over $\Q$ the embedding is an isomorphism, because the [[cyclotomic polynomials|0W0KVBM54S]] are [[irreducible over the rationals|0SK1VHZ5R2]].
The fields $\Q(\zeta_n)$ are the cyclotomic fields of number theory.

## Explanations

### L1 human

The $n$-th roots of unity are the $n$ complex numbers whose $n$-th power is $1$; they sit at the corners of a regular $n$-gon on the unit circle.
Adjoining one primitive root $\zeta_n$ gives all the others, because they are powers of it.
Each symmetry of $\Q(\zeta_n)$ sends $\zeta_n$ to a power $\zeta_n^k$ with $k$ coprime to $n$.
For $n = 4$, $\zeta_4 = i$ and the extension is $\Q(i)$, with the two symmetries $i \mapsto i$ and $i \mapsto -i$.