Ray class groups refine the ordinary ideal class group by remembering congruence conditions at selected finite primes and sign conditions at selected real embeddings.
The ordinary class group asks whether an ideal becomes principal. A ray class group asks a stricter question: if the ideal is principal, can it be generated by an element satisfying prescribed local congruences and positivity conditions? That extra arithmetic information is exactly what class field theory needs to organise abelian extensions.
This is Guide 41 in the Bukit Timah Tutor Computational Number Theory series. It extends the class-group algorithms of Guide 20, the prime-ideal arithmetic of Guide 38, and the S-unit viewpoint of Guide 40.
1. From class groups to ray class groups
For a number field K, the ideal class group is
Cl(K)=I_K/P_K
where I_K is the group of nonzero fractional ideals and P_K the subgroup of principal fractional ideals.
A ray class group modifies both numerator and denominator. We restrict to ideals coprime to a chosen finite modulus and quotient only by principal ideals generated by elements satisfying specified congruence and sign conditions.
2. A modulus
A modulus is written
m = m_0 m_∞
where m_0 is a nonzero integral ideal and m_∞ is a chosen set of real embeddings of K.
The finite part records congruence conditions. The infinite part records positivity conditions.
If K has no real embeddings, m_∞ is empty.
3. Ideals coprime to the modulus
Let I_m be the group of fractional ideals whose prime-ideal factorisation is coprime to m_0.
Equivalently, every prime ideal dividing m_0 must occur with valuation zero in the ideal.
This removes from the ray class group exactly the primes where the modulus is prescribing additional local behaviour.
4. Principal ray subgroup
Let P_(m,1) be the subgroup generated by principal ideals (α) such that
α ≡ 1 (mod m_0)
in the appropriate local/integral sense and
σ(α) > 0
for every real embedding σ contained in m_∞.
Then the ray class group is
Cl_m(K)=I_m/P_(m,1).
5. Why positivity appears
At a real place there are two connected components of R*: positive and negative numbers.
Including that real place in m_∞ forces generators of principal ray ideals to lie in the positive component.
This is the source of the narrow class group when all real places are included and the finite modulus is trivial.
6. Narrow class group
If m_0=(1) and m_∞ contains all real embeddings, the ray class group is the narrow class group.
It differs from the ordinary class group exactly when principal ideals may require generators with sign patterns that cannot be corrected by units.
For totally complex fields there is no difference because there are no real sign conditions.
7. The computational exact sequence
A central exact sequence has the shape
O_K^× → (O_K/m_0)^× × sign-data
→ Cl_m(K)
→ Cl(K)
→ 1.
The precise sign factor depends on m_∞.
Computationally this says: start with the ordinary class group, compute the finite residue-unit group and sign conditions, then quotient by the image of global units.
8. Why units matter
Two generators α and uα define the same principal ideal when u is a unit.
But α and uα may have different residue classes modulo m_0 or different signs at real places.
Therefore the unit group maps into the local congruence/sign group and must be divided out when constructing the ray class group.
This is why efficient unit-group computation is a prerequisite rather than an optional extra.
9. Worked example over Q
Take
K=Q
and modulus
m=5∞.
The ordinary class group of Q is trivial. There is one real place and the positivity condition is active.
The ray class group is naturally isomorphic to
(Z/5Z)^×.
Therefore
Cl_(5∞)(Q) ≅ C4.
10. Representatives modulo 5
The invertible residue classes are
1,2,3,4 mod5.
Multiplication modulo5 makes this a cyclic group of order4.
The class of2 generates it:
2,4,3,1.
In class-field language this ray class group will become the Galois group of Q(ζ_5)/Q.
11. Remove the infinite place
Now use only the finite modulus 5.
Since −1 is an allowed global unit and no positivity condition forbids it, the unit classes ±1 are identified.
Thus
Cl_5(Q) ≅ (Z/5Z)^×/{±1} ≅ C2.
This smaller ray class group corresponds to the maximal real subfield of Q(ζ_5), namely Q(√5).
12. Finite modulus powers
If the finite modulus contains P^n, the condition
α≡1 (mod P^n)
remembers progressively deeper local precision.
As n increases, the principal ray subgroup becomes smaller and the ray class group can grow.
This creates the arithmetic tower behind higher conductor abelian extensions.
13. Conductors
An abelian extension L/K has a conductor modulus f(L/K), the smallest modulus through which its Artin reciprocity map factors.
Ramified primes and their ramification depth contribute to the finite part. Real places that become complex may contribute to the infinite part.
Computing conductors is therefore the reverse problem of choosing a modulus: instead of asking what class field a modulus defines, ask what smallest modulus explains a given abelian extension.
14. Generalised ideal classes
Ordinary ideal classes identify two ideals when their quotient is any principal ideal.
Ray classes identify them only when the quotient is principal with a generator satisfying the ray conditions.
So the ray class group is a finer partition of ideals. It remembers arithmetic information deliberately discarded by the ordinary class group.
15. Relation matrices
Computationally, finitely generated abelian groups are handled with integer relation matrices.
Class-group relations, residue-unit generators, sign data and unit images can be assembled into one presentation matrix.
Smith normal form then converts the presentation into invariant factors
C_(n1) × ... × C_(nr).
This is the same exact integer linear algebra used throughout Guide18.
16. Computing (O_K/m_0)^×
Factor
m_0=∏P_i^(a_i).
Chinese remainder decomposition gives
O_K/m_0 ≅ ∏ O_K/P_i^(a_i).
The unit group decomposes correspondingly.
At prime powers, local methods and residue-field structure provide generators and relations.
17. Unit image modulo the modulus
Suppose ε_1,…,ε_r are fundamental units and μ_K is the torsion subgroup.
Reduce each generator modulo m_0 and record signs at the selected infinite places.
The subgroup generated by those images is exactly the global-unit ambiguity that must be removed from the local data.
18. Example of a ray equivalence test
Let ideals A and B be coprime to m_0.
To test whether they represent the same ray class, compute
A B^(-1).
First test whether this ideal is principal. If not, the ray classes differ.
If it is principal, say
AB^(-1)=(α),
then test whether α can be modified by a unit so that the required congruence and positivity conditions hold.
19. Ordinary class group as a quotient
There is a natural surjection
Cl_m(K) → Cl(K).
Forgetting ray conditions merges several ray classes into one ordinary ideal class.
The kernel measures the extra congruence/sign information introduced by the modulus.
20. Ray class group size
The ray class number can be computed from an exact sequence or ray class number formula involving:
ordinary class number residue-unit group size unit-image index infinite sign factors.
The formula is useful both for prediction and verification of a computed group presentation.
21. Prime ideals and ray classes
A prime ideal P not dividing the modulus determines a ray class [P].
In the corresponding ray class field, this class becomes the Frobenius/Artin element at P.
Thus a purely ideal-theoretic residue class turns into a Galois automorphism.
Guide42 develops this bridge.
22. Modulus design
Choosing m too small may exclude the desired extension because its conductor does not divide m.
Choosing m much too large creates a larger ray class group and more subgroups than necessary.
Efficient class-field computation therefore benefits from conductor bounds and local ramification information before the global group is constructed.
23. Software workflow
A reliable ray-class computation should record:
field K and maximal order O_K modulus m_0 and selected real places factorisation of m_0 ordinary class group unit group local residue-unit generators unit images presentation matrix Smith normal form map from ideals to ray classes.
24. Verification receipt
For every proposed generator relation, verify it inside the ideal group and local residue groups.
Then verify the invariant-factor product against the independently computed ray class number.
For sample ideals P not dividing the modulus, compute their ray coordinates and confirm orders predicted by the group structure.
25. Common mistakes
1. Forgetting to remove ideals dividing m_0 from the ray ideal group. 2. Treating α≡1 mod m_0 as an ordinary integer congruence when m_0 is a number-field ideal. 3. Ignoring infinite positivity conditions.
4. Forgetting to quotient by global units. 5. Confusing a ray class group with a ray class field. 6. Assuming the ordinary class group determines all abelian extensions. 7. Ignoring the conductor when comparing moduli. 8. Using a nonmaximal order as though it were O_K.
26. Practice set
1. Define a modulus. 2. What is I_m? 3. What conditions define P_(m,1)? 4. Define Cl_m(K).
5. What does the infinite part control? 6. What is the narrow class group? 7. Why do units enter the exact sequence? 8. Compute Cl_(5∞)(Q).
9. Compute Cl_5(Q). 10. What is a conductor? 11. How does Chinese remainder decomposition help? 12. What map connects prime ideals to class-field automorphisms?
27. Answers
1. A finite ideal part times a selected set of real places. 2. Fractional ideals coprime to the finite modulus. 3. Principal ideals generated by α congruent to1 mod m_0 and positive at the selected real places. 4. I_m/P_(m,1).
5. Positivity at real embeddings. 6. The ray class group with trivial finite modulus and all real places included. 7. Different generators of the same principal ideal can have different residues/signs. 8. (Z/5Z)^×≅C4.
9. (Z/5Z)^×/{±1}≅C2. 10. The minimal modulus through which the abelian extension’s reciprocity map factors. 11. It decomposes residue rings and their unit groups prime-power by prime-power. 12. The Artin reciprocity map.
Sources and further study
Henri Cohen, A Survey of Computational Class Field Theory, begins the computational class-field pipeline with ray class groups, unit groups and exact sequences of finite abelian groups. Standard computational algebraic-number-theory references develop modulus arithmetic, narrow conditions and ray class number formulas in detail.
Continue through Batch 11
Continue to Guide 42: Computational Class Field Theory, Guide 43: p-Adic Logarithms and Local Units, and Guide 44: PSLQ and Algebraic Number Recognition.