---
tag: AW00HPFB54
kind: statement
role: theorem
name: Kronecker–Weber theorem
msc: [11R20, 11R18, 11R37]
uses: [0QMZAQ6QQ4, 082GHNPJ53, NN77VYSGXR, 0YV2CKP2AV, 0C1Y04SZ1W, BEKCEYHCHX, D3HEJTVY7G, 7SV4WA92SM, KMVW45B0TT]
about: [NN77VYSGXR]
sources:
  - url: https://en.wikipedia.org/wiki/Kronecker%E2%80%93Weber_theorem
same_as:
  wikidata: Q1369453
  wikipedia: Kronecker–Weber theorem
---

## Statement

Let $L$ be a [[finite extension|0C1Y04SZ1W]] of $\Q$ that is a [[Galois extension|0QMZAQ6QQ4]] with [[abelian|082GHNPJ53]] Galois group.
Then there is a positive integer $n$ such that $L$ is contained in the [[cyclotomic field|NN77VYSGXR]] $\Q(\zeta_n)$.
Here $\zeta_n$ is a primitive $n$-th [[root of unity|0YV2CKP2AV]].

## Remarks

The converse holds as well: $\Gal(\Q(\zeta_n)/\Q) \cong (\Z/n\Z)^\times$ is abelian, so every subfield of $\Q(\zeta_n)$ is abelian over $\Q$.
One standard proof reduces to cyclic extensions of prime power degree and studies their ramification groups at the ramified primes.
A key input is that $\Q$ has no nontrivial extension unramified at all primes, a consequence of the [[Hermite–Minkowski theorem|KMVW45B0TT]].
Another proof comes from [[class field theory|BEKCEYHCHX]]: the ray class field of $\Q$ for the modulus $n\infty$ is $\Q(\zeta_n)$.

Example: every [[quadratic field|7SV4WA92SM]] is abelian over $\Q$, so it lies in a cyclotomic field.
For instance $\sqrt{2} = \zeta_8 + \zeta_8^{-1}$ lies in $\Q(\zeta_8)$, and $\sqrt{5} = \zeta_5 - \zeta_5^2 - \zeta_5^3 + \zeta_5^4$ lies in $\Q(\zeta_5)$.

The abelian hypothesis is needed.
The field $\Q(\sqrt[3]{2})$ is not contained in any $\Q(\zeta_n)$: it is not Galois over $\Q$, while every subfield of $\Q(\zeta_n)$ is Galois over $\Q$.

## Explanations

### L1 human

Roots of unity are the solutions of $x^n = 1$.
The theorem says that every number field with a commutative Galois group over $\Q$ can be built from roots of unity.
So the abelian extensions of $\Q$ have a complete and explicit description.

### L2 human

The maximal abelian extension of $\Q$ is the field generated by all roots of unity.
Its Galois group is $\widehat{\Z}^\times$, the unit group of the profinite completion $\widehat{\Z}$ of $\Z$.
Hilbert's twelfth problem asks for similar explicit generators for the abelian extensions of other number fields.
For imaginary quadratic fields this is achieved by [[complex multiplication|D3HEJTVY7G]], using values of elliptic and modular functions.