# Ray class field

A **ray class field** is canonically attached to a modulus: for a base number field \( K \) and a modulus \( \mathfrak{m} \), the ray class field of \( K \) modulo \( \mathfrak{m} \) is a finite abelian extension \( L/K \) for which the Artin map gives an isomorphism between the ray class group \( \mathrm{Cl}_K^{\mathfrak{m}} \) and \( \mathrm{Gal}(L/K) \), while for a global function field the full ray class field modulo a divisor is infinite.<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html)</sup><sup> • </sup><sup>[15](https://www.math.uzh.ch/sepp/magma-2.19.3-cr/html/text448.htm)</sup> Ray class fields organize all finite abelian extensions of a global field: every such extension is contained in some ray class field, and the smallest one containing it is measured by the conductor.<sup>[2](https://math.dartmouth.edu/~trs/expository-papers/CFT.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | For a modulus \( \mathfrak{m} \), the ray class field is the unique finite abelian extension \( L/K \) with defining modulus \( \mathfrak{m} \) such that the Artin map \( \psi_{L/K}^{\mathfrak{m}} \) is an isomorphism, equivalently \( N_{L/K}\mathrm{Cl}_L^{\mathfrak{m}} =  \)<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html)</sup> |
| Galois group | \( \mathrm{Cl}_{\mathfrak{m}}(K) \cong \mathrm{Gal}(K_{\mathfrak{m}}/K) \) via the Artin map, for \( K \) a number field<sup>[3](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup> |
| Hilbert class field | The ray class field for the trivial modulus \( (1) \); it is the maximal abelian extension unramified at all places, with \( \mathrm{Gal}(L/K) \cong \mathrm{Cl}_K \)<sup>[4](https://math.mit.edu/classes/18.785/2016fa/LectureNotes21.pdf)</sup> |
| Over \( \mathbb{Q} \) | The ray class field of \( \mathbb{Q} \) for \( (f)\infty \) is the cyclotomic field \( \mathbb{Q}(\mu_f) \); for \( (f) \) it is the maximal real subfield \( \mathbb{Q}(\zeta_f + \zeta_f^{-1}) \)<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html)</sup> |
| Conductor | For any finite abelian \( L/K \), the conductor \( \mathfrak{f}_{L/K} \) is the smallest modulus \( \mathfrak{m} \) with \( L \subseteq K_{\mathfrak{m}} \); a prime ramifies in \( L/K \) exactly when it divides the conductor<sup>[2](https://math.dartmouth.edu/~trs/expository-papers/CFT.pdf)</sup> |
| Existence theorem | Every subgroup \( H \) of \( \mathrm{Cl}_K^{\mathfrak{m}} \) over a number field equals \( N_{L/K}\mathrm{Cl}_L^{\mathfrak{m}} \) for a unique finite abelian extension \( L \) with defining modulus \( \mathfrak{m} \), while over a global function field the finite abelian extensions correspond to the finite-index subgroups<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html)</sup> |
| Function fields | For a global function field the ray class group is infinite, and the Artin map has dense rather than surjective image<sup>[3](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup> |

## The ray class group

The ray class field is defined through its [Galois group](https://www.edgepedia.org/galois-group). A modulus \( \mathfrak{m} = \mathfrak{m}_0\mathfrak{m}_{\infty} \) consists of an integral ideal \( \mathfrak{m}_0 \) of the ring of integers \( \mathcal{O}_K \) and a product \( \mathfrak{m}_{\infty} \) of real places.<sup>[5](https://planetmath.org/anexactsequenceforrayclassgroups)</sup> Let \( I_K^{(\mathfrak{m})} \) be the free abelian group of fractional ideals generated by the prime ideals not dividing \( \mathfrak{m} \), and let \( P_{\mathfrak{m}} \) be the subgroup of principal ideals \( (\alpha) \) with \( \alpha \equiv 1 \bmod \mathfrak{m} \) (positive at each real place in \( \mathfrak{m}_{\infty} \)). The ray class group is the quotient \( \mathrm{Cl}_{\mathfrak{m}}(K) = I_K^{(\mathfrak{m})}/P_{\mathfrak{m}} \).<sup>[3](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup>

For \( \mathfrak{m} = (1) \) this quotient is the ordinary ideal class group \( \mathrm{Cl}(K) \). If \( \mathfrak{m} \) is the product of all real places, the quotient is the narrow class group, generated by classes of totally positive elements.<sup>[3](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup> The ray class group is finite for a number field; for a global function field it is infinite, since the quotient contains a copy of \( q^{\mathbb{Z}} \).<sup>[3](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup> The units of \( K \) control the size of the ray class group through the exact sequence

\[ \mathcal{O}_K^{\times} \longrightarrow (\mathcal{O}_K/\mathfrak{m})^{\times} \longrightarrow \mathrm{Cl}(K, \mathfrak{m}) \longrightarrow \mathrm{Cl}(K) \longrightarrow 1, \]

where \( (\mathcal{O}_K/\mathfrak{m})^{\times} \cong (\mathcal{O}_K/\mathfrak{m}_0)^{\times} \times (\mathbb{Z}/2\mathbb{Z})^{|\mathfrak{m}_{\infty}|} \).<sup>[5](https://planetmath.org/anexactsequenceforrayclassgroups)</sup>

## Definition and existence

The existence theorem of global class field theory guarantees that ray class fields exist: for a number field \( K \) and a modulus \( \mathfrak{m} \), there is a unique finite abelian extension \( L \) with defining modulus \( \mathfrak{m} \) such that \( N_{L/K}\mathrm{Cl}_L^{\mathfrak{m}} = 1 \), and for this extension the Artin map \( \psi_{L/K}^{\mathfrak{m}} \) is an isomorphism.<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html)</sup> In the idele-theoretic formulation, \( K_{\mathfrak{m}} \) is the class field corresponding to the open subgroup \( K^{\times}U_{\mathfrak{m}}/K^{\times} \) of \( \mathbb{A}_K^{\times}/K^{\times} \).<sup>[3](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup>

Two structural consequences follow. First, every finite abelian extension of \( K \) for which \( \mathfrak{m} \) is a defining modulus is contained in the ray class field of modulus \( \mathfrak{m} \), so the ray class fields form a cofinal system of abelian extensions.<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html)</sup><sup> • </sup><sup>[3](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup>  Second, subfields of a ray class field correspond to quotients of the ray class group via Artin reciprocity.<sup>[4](https://math.mit.edu/classes/18.785/2016fa/LectureNotes21.pdf)</sup> Over \( \mathbb{Q} \) this recovers the [Kronecker–Weber theorem](https://www.edgepedia.org/kronecker-weber-theorem), that every finite abelian extension of \( \mathbb{Q} \) lies in a cyclotomic field.<sup>[2](https://math.dartmouth.edu/~trs/expository-papers/CFT.pdf)</sup>

## The conductor

Every finite abelian extension \( L/K \) has a conductor \( \mathfrak{f}_{L/K} \), the smallest modulus \( \mathfrak{m} \) such that \( L \subseteq K_{\mathfrak{m}} \).<sup>[2](https://math.dartmouth.edu/~trs/expository-papers/CFT.pdf)</sup> A prime of \( K \) ramifies in \( L \) exactly when it divides the conductor, so the conductor records both the ramification and the minimal ray class field containing \( L \).<sup>[2](https://math.dartmouth.edu/~trs/expository-papers/CFT.pdf)</sup> For \( L/K \) finite abelian, the finite part of the conductor factors as \( \mathfrak{f}_{L/K,f} = \prod_{\mathfrak{p}} (\mathfrak{f}_{L_{\mathfrak{P}}/K_{\mathfrak{p}}} \cap \mathcal{O}_K) \), a product of local conductors over the primes of \( K \).<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html)</sup> For example, the conductor of the ray class field \( \mathbb{Q}(\mu_{12}) \) of \( \mathbb{Q}(i) \) with modulus \( (3) \) is \( (3) \) itself, since \( (3) \) ramifies in \( \mathbb{Q}(\mu_{12})/\mathbb{Q}(i) \).<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html)</sup>

## The Hilbert class field

The ray class field for the trivial modulus \( (1) \) is the [Hilbert class field](https://www.edgepedia.org/hilbert-class-field), the maximal abelian extension of \( K \) unramified at all places, finite or infinite.<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html)</sup><sup> • </sup><sup>[4](https://math.mit.edu/classes/18.785/2016fa/LectureNotes21.pdf)</sup>  The reciprocity law shows that \( (1) \) is a defining modulus precisely for the finite abelian unramified extensions, so the ray class field of modulus \( (1) \) is the largest of them.<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html)</sup> Its Galois group is canonically isomorphic to the ideal class group \( \mathrm{Cl}_K \), and every ray class field of \( K \) contains the Hilbert class field, so a ray class field is a tower of two abelian extensions, the lower one independent of the modulus.<sup>[4](https://math.mit.edu/classes/18.785/2016fa/LectureNotes21.pdf)</sup>

The Hilbert class field carries arithmetic information about \( K \) itself: a nonzero prime ideal of \( \mathcal{O}_K \) is principal if and only if it splits completely in the Hilbert class field.<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html)</sup> Each fractional ideal of \( K \) becomes principal in the Hilbert class field, which motivates the class field tower problem, asking whether iterating the construction eventually reaches a field of class number one.<sup>[2](https://math.dartmouth.edu/~trs/expository-papers/CFT.pdf)</sup>

## Ray class fields over the rationals

For \( K = \mathbb{Q} \) the ray class fields are explicit. The ray class field for the modulus \( (f)\infty \) is the cyclotomic field \( \mathbb{Q}(\mu_f) \): only the places dividing \( (f)\infty \) ramify there, and the Artin map sends the class of a positive integer \( a \) coprime to \( f \) to the automorphism \( \sigma_a \) with \( \sigma_a(\zeta_f) = \zeta_f^a \).<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html)</sup> The ray class group is then \( \mathrm{Cl}_{\mathbb{Q}}^{(f)\infty} \cong (\mathbb{Z}/f\mathbb{Z})^{\times} \).<sup>[3](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup> Dropping the infinite place quotients out the element \( -1 \), and the ray class field for \( (f) \) is the maximal real subfield \( \mathbb{Q}(\zeta_f + \zeta_f^{-1}) \).<sup>[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html)</sup><sup> • </sup><sup>[5](https://planetmath.org/anexactsequenceforrayclassgroups)</sup>

The ray class group over \( \mathbb{Q} \) also recovers Dirichlet's theorem on primes in arithmetic progressions: the primes in each coset of \( I_K^{\mathfrak{m}} \) modulo the relevant subgroup have natural density \( 1/[I_K^{\mathfrak{m}} : H] \), which for \( \mathbb{Q} \) and \( \mathfrak{m} = m\infty \) is the uniform distribution of primes among residue classes modulo \( m \).<sup>[2](https://math.dartmouth.edu/~trs/expository-papers/CFT.pdf)</sup>

## History

The unramified case was formulated by [David Hilbert](https://www.edgepedia.org/david-hilbert), whose 1898 and 1899 work introduced the class field and the quadratic norm residue symbol, with existence proved for base fields of class number 2 and 4; Philipp Furtwängler later proved the existence of Hilbert class fields in general.<sup>[6](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B25/pdf/B25_008.pdf)</sup> The full theory, in which ray class fields for arbitrary moduli classify the abelian extensions, was proved by Teiji Takagi, who announced it at the International Congress of Mathematicians in [Strasbourg](https://www.edgepedia.org/strasbourg) in 1920 and published his main paper that year, generalizing Hilbert's class field; [Emil Artin](https://www.edgepedia.org/emil-artin)'s 1927 reciprocity law then identified the Galois group of each ray class field with its ray class group.<sup>[7](https://mathshistory.st-andrews.ac.uk/Biographies/Takagi/)</sup><sup> • </sup><sup>[2](https://math.dartmouth.edu/~trs/expository-papers/CFT.pdf)</sup>

## Computation and explicit construction

Ray class groups and fields are implemented in the standard computational algebra systems. Sage's `ray_class_number` method returns the order of the ray class group modulo an ideal, wrapping PARI's `bnrclassno`; for a prime above 3 in \( \mathbb{Q}(\sqrt{-23}) \) it returns 3, and for the ideal (5) in the cubic extension \( \mathbb{Q}(\sqrt{-23})(\sqrt[3]{z}) \) it returns 5184.<sup>[8](https://doc.sagemath.org/html/en/reference/number_fields/sage/rings/number_field/number_field_ideal.html)</sup> PARI/GP's `Buchray` initializes a ray class group structure from a bnf and a modulus, and functions such as `bnrconductor` and `bnrisconductor` compute and test conductors of class fields defined by subgroups.<sup>[9](https://pari.math.u-bordeaux.fr/dochtml/html-stable/usersch6.html)</sup> Magma's `RayClassField(D)` constructs the full ray class field modulo a divisor \( D \), an abelian extension unramified outside the support of \( D \) whose automorphism group is canonically isomorphic to the ray class group modulo \( D \); its documentation demonstrates the construction by computing the 12th cyclotomic field, whose ray class group modulo \( (12, \infty) \) is \( \mathbb{Z}/2 \oplus \mathbb{Z}/2 \).<sup>[10](https://docs.magma-maths.org/GlobalFields/ClassFieldTheory/creation.html)</sup> Hecke.jl likewise constructs ray class fields from quotients of ray class groups, including the \( n \)-th cyclotomic field as a ray class field.<sup>[11](https://docs.hecke.thofma.com/dev/manual/number_fields/class_fields)</sup>

Explicit generation of ray class fields, in the sense of producing the field by special values of analytic functions, is [Hilbert's twelfth problem](https://www.edgepedia.org/hilberts-twelfth-problem). Known cases include the cyclotomic (Kronecker–Weber) context, the real-cyclotomic context, where the periodic locus of the Chebyshev \( \Lambda \)-structure on \( \mathbb{A}^1 \) generates the real ray class fields \( \mathbb{Q}(\zeta_n + \zeta_n^{-1}) \), and imaginary quadratic fields, where special values of modular functions and theta constants generate the ray class fields.<sup>[12](https://maths-people.anu.edu.au/borger/papers/_all/ExplicitClassFieldTheory.pdf)</sup><sup> • </sup><sup>[13](https://arxiv.org/abs/1203.4662)</sup>

## Function fields

The theory extends to global function fields, with one structural difference: the ray class group modulo a divisor is infinite, and the Artin map induces a map with dense image rather than an isomorphism onto the Galois group.<sup>[3](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup> Ray class fields of function fields appear in applications such as algebraic geometry codes, where subfields with cyclic Galois group of narrow ray class fields underlie list-decodable codes whose error fraction approaches \( 1 - R \) for rate \( R \).<sup>[14](https://arxiv.org/abs/1302.6660)</sup>

## References

1. [Romyar Sharifi, Algebraic Number Theory, Chapter 7: Global class field theory via ideals, UCLA](https://www.math.ucla.edu/~sharifi/notes/algnum-ch07.html)
2. [An Overview of Class Field Theory, Dartmouth College expository paper](https://math.dartmouth.edu/~trs/expository-papers/CFT.pdf)
3. [Li Chung, Class field theory lecture notes, Columbia University](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)
4. [18.785 Number Theory I, Lecture 21: Ray class fields and Artin reciprocity, MIT (2016)](https://math.mit.edu/classes/18.785/2016fa/LectureNotes21.pdf)
5. [An exact sequence for ray class groups, PlanetMath](https://planetmath.org/anexactsequenceforrayclassgroups)
6. [Takagi's Class Field Theory: From where? and to where?, RIMS, Kyoto University](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B25/pdf/B25_008.pdf)
7. [Teiji Takagi (1875–1960), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Takagi/)
8. [Ideals of number fields, SageMath reference manual](https://doc.sagemath.org/html/en/reference/number_fields/sage/rings/number_field/number_field_ideal.html)
9. [PARI/GP Manual, part usersch6](https://pari.math.u-bordeaux.fr/dochtml/html-stable/usersch6.html)
10. [Creation, Class Field Theory, Magma Handbook](https://docs.magma-maths.org/GlobalFields/ClassFieldTheory/creation.html)
11. [Class Field Theory, Hecke.jl documentation](https://docs.hecke.thofma.com/dev/manual/number_fields/class_fields)
12. [Borger and de Smit, Explicit class field theory and the algebraic geometry of Λ-rings](https://maths-people.anu.edu.au/borger/papers/_all/ExplicitClassFieldTheory.pdf)
13. [Koo and Yoon, Construction of class fields over cyclotomic fields, arXiv:1203.4662](https://arxiv.org/abs/1203.4662)
14. [Optimal rate algebraic list decoding using narrow ray class fields, arXiv:1302.6660](https://arxiv.org/abs/1302.6660)
15. [Properties of Class Fields](https://www.math.uzh.ch/sepp/magma-2.19.3-cr/html/text448.htm)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Algebraic number theory › Class field theory › Global class field theory*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license