# Kronecker–Weber theorem

The **Kronecker–Weber theorem** states that every finite abelian extension of the rational numbers \(\mathbb{Q}\) is contained in a cyclotomic field \(\mathbb{Q}(\zeta_m)\), the field obtained by adjoining a primitive \(m\)-th [root of unity](https://www.edgepedia.org/root-of-unity) \(\zeta_m\) to \(\mathbb{Q}\)<sup>[1](https://ocw.mit.edu/courses/18-785-number-theory-i-fall-2021/mit18_785f21_full_lec.pdf)</sup>. An abelian extension is a [Galois extension](https://www.edgepedia.org/galois-extension) whose [Galois group](https://www.edgepedia.org/galois-group) is abelian; the cyclotomic field \(\mathbb{Q}(\zeta_n)\) is itself such an extension, with Galois group \((\mathbb{Z}/n\mathbb{Z})^*\), and the theorem says that these fields and their subfields exhaust all examples<sup>[2](https://kskedlaya.org/cft/sec_kronweb.html)</sup>. The result is one of the earliest theorems of class field theory and is sometimes called the Kronecker–Weber–Hilbert theorem<sup>[3](https://mathworld.wolfram.com/Kronecker-WeberTheorem.html)</sup>.

| Key fact | Detail |
|---|---|
| Statement | Every finite abelian extension of \(\mathbb{Q}\) lies in a cyclotomic field \(\mathbb{Q}(\zeta_m)\)<sup>[1](https://ocw.mit.edu/courses/18-785-number-theory-i-fall-2021/mit18_785f21_full_lec.pdf)</sup> |
| History | Stated by Kronecker in 1853 with a partial proof for odd-degree extensions; Weber's 1886 proof has gaps; Hilbert gave the first complete proof in 1896<sup>[1](https://ocw.mit.edu/courses/18-785-number-theory-i-fall-2021/mit18_785f21_full_lec.pdf)</sup><sup> • </sup><sup>[3](https://mathworld.wolfram.com/Kronecker-WeberTheorem.html)</sup> |
| Local version | Every finite abelian extension of \(\mathbb{Q}_p\) lies in \(\mathbb{Q}_p(\zeta_m)\); the local theorem implies the global one<sup>[1](https://ocw.mit.edu/courses/18-785-number-theory-i-fall-2021/mit18_785f21_full_lec.pdf)</sup> |
| Conductor | The smallest \(n\) with \(K \subseteq \mathbb{Q}(\zeta_n)\) is the conductor of \(K/\mathbb{Q}\), which governs how primes of \(\mathbb{Q}\) split in \(K\)<sup>[2](https://kskedlaya.org/cft/sec_kronweb.html)</sup> |
| Zeta factorization | For an abelian number field \(K\), the Dedekind zeta function factors as \(\zeta_K(s) = \prod_{\chi} L(\chi, s)\) over the associated Dirichlet characters<sup>[4](https://ma.huji.ac.il/~deshalit/new_site/files/NTcourse.pdf)</sup> |
| Maximal abelian extension | The maximal abelian extension of \(\mathbb{Q}\) is the union of all cyclotomic fields \(\mathbb{Q}(\zeta_n)\)<sup>[5](https://citeseerx.ist.psu.edu/document?doi=85577a85b5e840b68fc3b1e583aaec77840d2db0&repid=rep1&type=pdf)</sup> |
| Generalization | Hilbert's twelfth problem asks for analogs for other base fields; it remains open in general<sup>[6](https://web.stanford.edu/~lindrew/18.784p.pdf)</sup> |

## Statement and meaning

A finite abelian extension \(K/\mathbb{Q}\) is a Galois extension of finite degree whose Galois group \(\mathrm{Gal}(K/\mathbb{Q})\) is abelian. The theorem classifies these extensions completely: each one sits inside some \(\mathbb{Q}(\zeta_m)\), and conversely every subfield of a cyclotomic field is abelian over \(\mathbb{Q}\)<sup>[2](https://kskedlaya.org/cft/sec_kronweb.html)</sup>. Equivalently, the maximal abelian extension of \(\mathbb{Q}\) inside a fixed algebraic closure is the union of all the cyclotomic fields<sup>[5](https://citeseerx.ist.psu.edu/document?doi=85577a85b5e840b68fc3b1e583aaec77840d2db0&repid=rep1&type=pdf)</sup>.

The theorem also has a computational reading: every [algebraic integer](https://www.edgepedia.org/algebraic-integer) whose Galois group is abelian can be expressed as a sum of \(n\)-th roots of unity for some \(n\)<sup>[3](https://mathworld.wolfram.com/Kronecker-WeberTheorem.html)</sup>.

## The conductor

The smallest positive integer \(n\) such that \(K \subseteq \mathbb{Q}(\zeta_n)\) is called the conductor of \(K/\mathbb{Q}\)<sup>[2](https://kskedlaya.org/cft/sec_kronweb.html)</sup>. It measures how far into the cyclotomic tower one must climb to reach \(K\), and it controls the splitting behavior of rational primes in \(K\): the Frobenius class of a prime unramified in \(K\) is determined by a congruence condition modulo the conductor<sup>[2](https://kskedlaya.org/cft/sec_kronweb.html)</sup>.

The conductor also enters analysis. The characters of \(\mathrm{Gal}(K/\mathbb{Q})\) can be identified with Dirichlet characters modulo the conductor, and the [Dedekind zeta function](https://www.edgepedia.org/dedekind-zeta-function) of \(K\) factors as

\[
\zeta_K(s) = \prod_{\chi \in G_K} L(\chi, s),
\]

a product of Dirichlet \(L\)-functions over the associated character group<sup>[4](https://ma.huji.ac.il/~deshalit/new_site/files/NTcourse.pdf)</sup>. This factorization is a consequence of the Kronecker–Weber theorem, since it requires knowing that \(K\) is a subfield of some \(\mathbb{Q}(\zeta_m)\)<sup>[1](https://ocw.mit.edu/courses/18-785-number-theory-i-fall-2021/mit18_785f21_full_lec.pdf)</sup>.

## History of the proofs

Kronecker stated the theorem in 1853 and provided a partial proof covering extensions of odd degree<sup>[1](https://ocw.mit.edu/courses/18-785-number-theory-i-fall-2021/mit18_785f21_full_lec.pdf)</sup>. Weber published a proof in 1886 that was believed to handle the remaining cases, but his argument contains gaps, as later commentators noted<sup>[1](https://ocw.mit.edu/courses/18-785-number-theory-i-fall-2021/mit18_785f21_full_lec.pdf)</sup>. Hilbert gave the first complete proof in 1896<sup>[3](https://mathworld.wolfram.com/Kronecker-WeberTheorem.html)</sup>, and Hilbert himself distinguished between Kronecker, who "stated" the theorem, and Weber, who proved it<sup>[7](https://www.researchgate.net/publication/237262060_On_the_History_of_Hilbert%27s_Twelfth_Problem_A_Comedy_of_Errors)</sup>. A common feature of the proofs is that, after "twisting" by suitable cyclotomic extensions, an abelian extension of \(\mathbb{Q}\) becomes unramified at all finite primes, hence has discriminant 1 and equals \(\mathbb{Q}\)<sup>[8](https://websites.math.leidenuniv.nl/algebra/localfields.pdf)</sup>.

Modern proofs usually run through class field theory, of which the theorem is a direct corollary, though it is also used historically as motivation for the general theory<sup>[8](https://websites.math.leidenuniv.nl/algebra/localfields.pdf)</sup>. Elementary proofs exist as well. One standard route reduces the general statement in several steps<sup>[9](https://warwick.ac.uk/fac/sci/maths/people/staff/sheth/kw.pdf)</sup>:

1. It suffices to treat cyclic extensions of prime-power degree, since a finite abelian Galois group decomposes as a product of cyclic groups of prime-power order and \(K\) is the compositum of the corresponding fixed fields<sup>[9](https://warwick.ac.uk/fac/sci/maths/people/staff/sheth/kw.pdf)</sup><sup> • </sup><sup>[10](https://math.berkeley.edu/~ogus/Math_254-07/KronWeb.pdf)</sup>.
2. It further suffices to treat the case where a single prime \(p\) is the only prime ramifying in \(K\), splitting into the cases \(p\) odd and \(p = 2\)<sup>[9](https://warwick.ac.uk/fac/sci/maths/people/staff/sheth/kw.pdf)</sup>.
3. The case \(p = 2\) is handled by induction on the degree, with the quadratic case as the base step<sup>[9](https://warwick.ac.uk/fac/sci/maths/people/staff/sheth/kw.pdf)</sup>.

The quadratic base case is itself classical: every quadratic extension of \(\mathbb{Q}\) lies in a cyclotomic field, a fact known to Gauss through what are now called Gauss sums, and it forms the basis of one of his proofs of quadratic reciprocity<sup>[2](https://kskedlaya.org/cft/sec_kronweb.html)</sup>.

## The local theorem

There is a local analog: every finite abelian extension of the \(p\)-adic field \(\mathbb{Q}_p\) lies in a cyclotomic field \(\mathbb{Q}_p(\zeta_m)\)<sup>[1](https://ocw.mit.edu/courses/18-785-number-theory-i-fall-2021/mit18_785f21_full_lec.pdf)</sup>. The local result implies the global one. For a finite abelian \(K/\mathbb{Q}\), each completion \(K_{\mathfrak{p}}\) at a prime over a ramified rational prime \(p\) is an abelian extension of \(\mathbb{Q}_p\), hence contained in some \(\mathbb{Q}_p(\zeta_{m_p})\); taking \(m\) as the product over ramified primes of the relevant prime-power parts shows \(K \subseteq \mathbb{Q}(\zeta_m)\)<sup>[1](https://ocw.mit.edu/courses/18-785-number-theory-i-fall-2021/mit18_785f21_full_lec.pdf)</sup><sup> • </sup><sup>[2](https://kskedlaya.org/cft/sec_kronweb.html)</sup>. The local theorem also holds for the archimedean completion \(\mathbb{Q}_\infty = \mathbb{R}\), in an uninteresting way<sup>[8](https://websites.math.leidenuniv.nl/algebra/localfields.pdf)</sup>.

An explicit form of the local theorem was proved by Jochen Koenigsmann, a mathematician at the [University of Oxford](https://www.edgepedia.org/university-of-oxford), and Benedikt Stock in 2022: if \(F/\mathbb{Q}_p\) is abelian of degree \(n = p^l \cdot m\) with \((p, m) = 1\), then

\[
F \subseteq \mathbb{Q}_p\bigl(\zeta_{p^{l+2}},\ \zeta_{p^n - 1}\bigr).
\]

Their proof is elementary and self-contained, using no tools beyond a second undergraduate algebra course apart from discrete valuation theory, and avoiding local class field theory and [Galois cohomology](https://www.edgepedia.org/galois-cohomology)<sup>[11](https://arxiv.org/abs/2206.05801)</sup>. The proof splits into a tame case, where the degree is prime to \(p\), and a wild case of \(p\)-power degree, treated separately and then combined<sup>[11](https://arxiv.org/abs/2206.05801)</sup>.

## Place in class field theory

The Kronecker–Weber theorem is the prototype of class field theory: it characterizes the abelian extensions of \(\mathbb{Q}\) by cyclotomic fields, just as class field theory characterizes abelian extensions of a general number field by ray class fields<sup>[12](https://androma.org/page/Solvability%20by%20Radicals)</sup>. For \(\mathbb{Q}\), the ray class field of modulus \(n\cdot\infty\) is exactly \(\mathbb{Q}(\zeta_n)\), and the ray class field of modulus \(n\) is its maximal real subfield \(\mathbb{Q}(\zeta_n + \zeta_n^{-1})\)<sup>[8](https://websites.math.leidenuniv.nl/algebra/localfields.pdf)</sup>.

The theorem also has consequences for how primes split. Assuming it, one deduces strong results about the splitting of primes of \(\mathbb{Q}\) in an abelian extension, governed by congruence conditions modulo the conductor<sup>[2](https://kskedlaya.org/cft/sec_kronweb.html)</sup>. The [Artin reciprocity law](https://www.edgepedia.org/artin-reciprocity-law) extends this phenomenon to abelian extensions of any number field<sup>[2](https://kskedlaya.org/cft/sec_kronweb.html)</sup>.

## Hilbert's twelfth problem and generalizations

[Hilbert's twelfth problem](https://www.edgepedia.org/hilberts-twelfth-problem) asks for generalizations of the Kronecker–Weber theorem to base fields other than \(\mathbb{Q}\), and for the analogs of the roots of unity in those fields<sup>[13](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/ComplexMultiplicationEC.pdf)</sup>. The problem remains open in general. The theory of complex multiplication resolves the case of imaginary quadratic fields \(\mathbb{Q}[\sqrt{d}]\) with \(d\) a squarefree negative integer, where torsion points of elliptic curves with complex multiplication play the role that roots of unity play for \(\mathbb{Q}\)<sup>[6](https://web.stanford.edu/~lindrew/18.784p.pdf)</sup>. For any number field other than \(\mathbb{Q}\), there are abelian extensions that are not cyclotomic, so no direct analog of the theorem's statement can hold<sup>[8](https://websites.math.leidenuniv.nl/algebra/localfields.pdf)</sup>.

Analogues have also been developed in other settings: a combinatorial proof of a Kronecker–Weber theorem for global fields of positive characteristic compares cyclic extensions of bounded conductor with subextensions of cyclotomic function fields<sup>[14](https://arxiv.org/abs/1307.3590)</sup>, and analogs exist for rational function fields over ultra-finite fields<sup>[15](https://arxiv.org/abs/2408.02158)</sup>.

## References

1. [18.785 Number Theory I, Lecture 20: The Kronecker-Weber theorem, MIT OpenCourseWare (Fall 2021)](https://ocw.mit.edu/courses/18-785-number-theory-i-fall-2021/mit18_785f21_full_lec.pdf)
2. [K. S. Kedlaya, "The Kronecker-Weber theorem," Section 1.1 of class field theory notes](https://kskedlaya.org/cft/sec_kronweb.html)
3. [Kronecker-Weber Theorem, Wolfram MathWorld](https://mathworld.wolfram.com/Kronecker-WeberTheorem.html)
4. [Number theory course notes, Hebrew University of Jerusalem (D. DeShalit)](https://ma.huji.ac.il/~deshalit/new_site/files/NTcourse.pdf)
5. [A combinatorial proof of the Kronecker–Weber theorem in positive characteristic (Citeseerx copy)](https://citeseerx.ist.psu.edu/document?doi=85577a85b5e840b68fc3b1e583aaec77840d2db0&repid=rep1&type=pdf)
6. [18.784 seminar paper on Hilbert's twelfth problem, Stanford University](https://web.stanford.edu/~lindrew/18.784p.pdf)
7. [On the History of Hilbert's Twelfth Problem: A Comedy of Errors, ResearchGate](https://www.researchgate.net/publication/237262060_On_the_History_of_Hilbert%27s_Twelfth_Problem_A_Comedy_of_Errors)
8. [Algebraic number theory / local fields lecture notes, Leiden University](https://websites.math.leidenuniv.nl/algebra/localfields.pdf)
9. [Proof of the Kronecker-Weber Theorem, University of Warwick (R. Sheth)](https://warwick.ac.uk/fac/sci/maths/people/staff/sheth/kw.pdf)
10. [The Kronecker-Weber Theorem, Math 254 notes, University of California, Berkeley (K. Oguiso course notes)](https://math.berkeley.edu/~ogus/Math_254-07/KronWeb.pdf)
11. [J. Koenigsmann, B. Stock (2022). An Elementary Proof of the Local Kronecker-Weber Theorem. arXiv.](https://arxiv.org/abs/2206.05801)
12. [Solvability by Radicals, Androma reference page](https://androma.org/page/Solvability%20by%20Radicals)
13. [Kronecker–Weber and complex multiplication, M. Waldschmidt, Sorbonne Université](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/ComplexMultiplicationEC.pdf)
14. [A combinatorial proof of the Kronecker--Weber Theorem in positive characteristic, arXiv](https://arxiv.org/abs/1307.3590)
15. [Ultra-Galois theory and an analogue of the Kronecker--Weber theorem for rational function fields over ultra-finite fields, arXiv](https://arxiv.org/abs/2408.02158)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Algebraic number theory › Class field theory › Explicit abelian extensions*

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