# Hilbert class field
The Hilbert class field of a number field K is the maximal unramified abelian extension of K, that is, the largest finite Galois extension of K with abelian [Galois group](https://www.edgepedia.org/galois-group) in which no prime of K, finite or infinite, ramifies.[1](https://mathworld.wolfram.com/HilbertClassField.html) • [2](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/chenal.pdf) It is unique and contains every other unramified abelian extension of K.[1](https://mathworld.wolfram.com/HilbertClassField.html) The field is the central example of class field theory, the theory that describes all abelian extensions of a number field in terms of data intrinsic to the field itself; as Kiran Kedlaya's notes put it, class field theory is literally "class field" theory, the theory of fields such as the Hilbert class field.[3](https://kskedlaya.org/cft/sec_hilbert.html)
| Key fact | Detail |
|---|---|
| Definition | The unique maximal unramified abelian extension L of a number field K, containing all other unramified abelian extensions of K[1](https://mathworld.wolfram.com/HilbertClassField.html) |
| Galois group | Gal(L/K) is isomorphic to the ideal class group of K, via the Artin reciprocity law[1](https://mathworld.wolfram.com/HilbertClassField.html) • [3](https://kskedlaya.org/cft/sec_hilbert.html) |
| Degree | [L : K] equals the class number of K[1](https://mathworld.wolfram.com/HilbertClassField.html) |
| Splitting criterion | A prime ideal of K splits completely in L if and only if it is principal[2](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/chenal.pdf) |
| Principal ideal theorem | Every ideal of K becomes principal in L (Furtwängler's theorem)[4](https://kskedlaya.org/cft/sec_principal.html) |
| Imaginary quadratic case | For K imaginary quadratic, K(j(E)) is the Hilbert class field for any elliptic curve E with complex multiplication by the ring of integers of K[5](https://www.math.ucla.edu/~jas/writings/cm-elliptic-curves-cft.pdf) |
| Iteration | Iterating the construction produces a class field tower[1](https://mathworld.wolfram.com/HilbertClassField.html) |
## Definition and basic properties
An extension K ⊂ L is abelian when it is Galois and Gal(L/K) is an abelian group. Ramification is tested at all primes, finite and infinite: an infinite prime of K is an embedding of K into the real or complex numbers, and it ramifies in L when a real embedding of K extends to a complex embedding of L. The Hilbert class field is unramified at every prime in this sense.[2](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/chenal.pdf)
The restriction is strong, yet a field can still have [unramified extensions](https://www.edgepedia.org/unramified-extension) of arbitrarily high degree; the thesis of Chenal gives K = Q(√(−2·3·5·7·11·13)) as an example.[2](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/chenal.pdf) A theorem going back to the foundations of class field theory guarantees that for any number field K there is a finite Galois extension L that is unramified and abelian over K and contains every unramified abelian extension of K; this L is the Hilbert class field, and it is unique.[2](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/chenal.pdf)
## The class group as Galois group
The substance of the construction is the identification of the Galois group with arithmetic of K itself. The [Artin reciprocity law](https://www.edgepedia.org/artin-reciprocity-law) gives a canonical isomorphism between the ideal class group Cl(K) and Gal(L/K).[3](https://kskedlaya.org/cft/sec_hilbert.html) Concretely, for an unramified abelian extension the Artin symbol is defined for every prime ideal of K, and the Artin map from the group of fractional ideals of K to Gal(L/K) is surjective with kernel exactly the subgroup of principal fractional ideals; it therefore induces an isomorphism C(O_K) ≅ Gal(L/K).[2](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/chenal.pdf) This appearance of the class group is why L is called a "class field".[2](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/chenal.pdf)
[Galois theory](https://www.edgepedia.org/galois-theory) then classifies all unramified abelian extensions of K: they correspond one to one to subgroups of the ideal class group, and if M/K corresponds to a subgroup H, the Artin map identifies Cl(K)/H with Gal(M/K).[2](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/chenal.pdf) In particular the degree [L : K] equals the class number of K.[1](https://mathworld.wolfram.com/HilbertClassField.html)
The correspondence also yields a concrete splitting law. A prime ideal of K splits completely in the Hilbert class field if and only if it is principal.[2](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/chenal.pdf) This criterion is a working tool: knowing the Hilbert class field lets one decide which primes are represented by which ideal classes, for example primes of the form x² + ny².[6](https://web.math.princeton.edu/~kk2703/kallal_cft_notes.pdf)
## The principal ideal theorem
A theorem of Philipp Furtwängler, a student of Hilbert, states that every ideal of K becomes principal when extended to the Hilbert class field L; the older vocabulary calls this "capitulation", from the former use of "capitulate" meaning "to become principal".[4](https://kskedlaya.org/cft/sec_principal.html) The proof applies Artin reciprocity to reduce the statement to a purely group-theoretic fact about the vanishing of a transfer homomorphism between Galois groups.[4](https://kskedlaya.org/cft/sec_principal.html)
A small example shows the phenomenon. For K = Q(√−5), the Hilbert class field is L = Q(√−5, √−1), and the nonprincipal ideal class represented by (2, 1 + √−5) is generated by the single element 1 + √−1 in L.[4](https://kskedlaya.org/cft/sec_principal.html) The property of making ideals principal is not specific to the Hilbert class field; other extensions can have it as well.[4](https://kskedlaya.org/cft/sec_principal.html)
## Explicit construction for imaginary quadratic fields
For general K the Hilbert class field is guaranteed to exist but is not given by a formula. For imaginary quadratic fields, the theory of complex multiplication supplies one. There is an intimate connection between the ideal class group of an imaginary quadratic field K and the set of elliptic curves with complex multiplication by the ring of integers O_K, tracked by their j-invariants; by class field theory these curves must say something about the Hilbert class field, and indeed K(j(E)) is exactly the Hilbert class field of K for E with complex multiplication by O_K.[5](https://www.math.ucla.edu/~jas/writings/cm-elliptic-curves-cft.pdf)
The same theory produces the Hilbert class polynomial. For K = Q(√D) with D a negative fundamental discriminant, adjoining j([a, b, c]) for any reduced quadratic form [a, b, c] of discriminant D gives the Hilbert class field, and the minimal polynomial H_D(X) = ∏ (X − j([a, b, c])) over the class group C(D) has integer coefficients; the Hilbert class field is precisely the splitting field of H_D(X).[2](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/chenal.pdf) When K has class number 1, the j-invariant is rational; for instance j(i) = 1728 and j(e^{πi/3}) = 0, and in general the j-invariant of a curve with complex multiplication by O_K is an algebraic integer.[5](https://www.math.ucla.edu/~jas/writings/cm-elliptic-curves-cft.pdf) Algorithms for computing H_D, including p-adic lifting and Chinese remainder methods using the class group action on CM curves over finite fields, are an active computational topic.[7](https://arxiv.org/abs/0802.0979)
## Class field towers and variants
Iterating the construction, taking the Hilbert class field of the Hilbert class field and so on, produces a class field tower.[1](https://mathworld.wolfram.com/HilbertClassField.html) A related variant is the narrow Hilbert class field H⁺, defined as the [ray class field](https://www.edgepedia.org/ray-class-field) for the modulus equal to the product of all real places of K; it corresponds to the narrow class group, which keeps track of signs of generators of principal ideals.[8](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html) For a totally real field F of degree n, the Hilbert class field is a totally real field of absolute degree n·h(R), where h(R) is the relevant class number, with the same root discriminant as F.[9](https://onlinelibrary.wiley.com/doi/10.1112/tlm3.12019)
## Place in Hilbert's twelfth problem
[Hilbert's twelfth problem](https://www.edgepedia.org/hilberts-twelfth-problem) asks for an explicit construction of the abelian extensions of a number field by analytic data intrinsic to the field.[10](https://www.math.tifr.res.in/conferences/ic2020/webpages/HTML/talks.html) The Kronecker–Weber theorem settles the case K = Q: the maximal abelian extension Q^ab is obtained by adjoining all roots of unity, which are the special values of the exponential function e^{2πix} at rational arguments.[11](https://sites.math.duke.edu/~dasgupta/papers/Integral-Gross-Stark.pdf) [Complex multiplication](https://www.edgepedia.org/complex-multiplication) gives the analogous statement for imaginary quadratic fields: F^ab is generated by j(E) and the Weber function values w(E[n]) on the torsion of an elliptic curve E with complex multiplication by O_F.[11](https://sites.math.duke.edu/~dasgupta/papers/Integral-Gross-Stark.pdf) Hilbert's original statement suggested that special values of elliptic modular functions generate the abelian extensions of imaginary quadratic fields, a claim that is not correct as stated, and the explicit theory of these fields took its modern shape only at the end of the 1920s after a series of corrections to work influenced by Hilbert's formulation.[12](https://smf.emath.fr/sites/default/files/2018-11/smf_sem-cong_3_243-273__sample.pdf)
For totally real fields the problem remained open much longer. Samit Dasgupta, professor of mathematics at [Duke University](https://www.edgepedia.org/duke-university), and Mahesh Kakde of the [Indian Institute of Science](https://www.edgepedia.org/indian-institute-of-science) proved the Brumer–Stark conjecture and, building on it, showed that the Brumer–Stark units together with n − 1 explicitly described square roots of elements of F generate the maximal abelian extension of any totally real field F of degree n, giving an unconditional construction that uses p-adic integration for infinitely many primes p; this is regarded as a p-adic solution of Hilbert's twelfth problem for totally real fields.[11](https://sites.math.duke.edu/~dasgupta/papers/Integral-Gross-Stark.pdf) • [13](https://sites.math.duke.edu/~dasgupta/papers/ICM.pdf) Their formulas generate narrow Hilbert class fields of real quadratic fields as a special case, with extensive tables of computations.[11](https://sites.math.duke.edu/~dasgupta/papers/Integral-Gross-Stark.pdf)
## References
1. [Hilbert Class Field, Wolfram MathWorld](https://mathworld.wolfram.com/HilbertClassField.html)
2. [The Hilbert Class Field (thesis chapter), Université de Bordeaux ALGANT](https://www.math.u-bordeaux.fr/~ybilu/algant/documents/theses/chenal.pdf)
3. [The Hilbert class field, Kedlaya, Class Field Theory notes](https://kskedlaya.org/cft/sec_hilbert.html)
4. [The principal ideal theorem, Kedlaya, Class Field Theory notes](https://kskedlaya.org/cft/sec_principal.html)
5. [Applications of complex multiplication to class field theory, UCLA](https://www.math.ucla.edu/~jas/writings/cm-elliptic-curves-cft.pdf)
6. [Class Field Theory notes, K. Prasad, Princeton](https://web.math.princeton.edu/~kk2703/kallal_cft_notes.pdf)
7. [Computing Hilbert Class Polynomials, arXiv:0802.0979](https://arxiv.org/abs/0802.0979)
8. [Class field theory, Liang-Chung Hsia / Columbia course notes](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)
9. [Definite orders with locally free cancellation, Smertnig, Trans. London Math. Soc. (2019)](https://onlinelibrary.wiley.com/doi/10.1112/tlm3.12019)
10. [Talk abstract: explicit class field theory and Stark's conjectures, TIFR](https://www.math.tifr.res.in/conferences/ic2020/webpages/HTML/talks.html)
11. [Dasgupta, S., Kakde, M. Brumer–Stark Units and Explicit Class Field Theory](https://sites.math.duke.edu/~dasgupta/papers/Integral-Gross-Stark.pdf)
12. [On the history of Hilbert's twelfth problem, SMF Séminaires et Congrès](https://smf.emath.fr/sites/default/files/2018-11/smf_sem-cong_3_243-273__sample.pdf)
13. [Dasgupta, S., Kakde, M. On the Brumer–Stark Conjecture and Refinements, ICM 2022](https://sites.math.duke.edu/~dasgupta/papers/ICM.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Algebraic number theory › Class field theory › Hilbert class theory and norm theorems*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Oct 15, 2026; Oct 25, 2026 · Last review: Oct 15, 2026*
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