Class field theory turns ideal classes into Galois automorphisms. Its central computational promise is that finite abelian extensions of a number field can be organised by ray class groups, conductors and reciprocity maps.
For computation, the theory has two directions. In the forward direction, start with a modulus and a subgroup of a ray class group and construct the corresponding abelian extension. In the reverse direction, start with a known abelian extension and compute its conductor and Artin map.
This is Guide 42 in the Bukit Timah Tutor Computational Number Theory series. It builds directly on Guide 41: Ray Class Groups.
1. The reciprocity statement
Let L/K be a finite abelian extension with conductor dividing a modulus m.
Class field theory gives a surjective Artin map
Art_m : Cl_m(K) → Gal(L/K)
whose kernel H determines the extension:
Gal(L/K) ≅ Cl_m(K)/H.
Conversely, suitable subgroups H of ray class groups correspond to finite abelian extensions.
2. Frobenius at an unramified prime
Let P be a prime ideal of K unramified in L and let Q lie above P.
The Frobenius automorphism is characterised on the residue field by
Frob_P(x) ≡ x^(N(P)) (mod Q).
Because L/K is abelian, this automorphism does not depend on the choice of Q.
The Artin map sends the ray class of P to Frob_P.
3. Multiplicativity
If an ideal coprime to the conductor factors as
a=∏P_i^(n_i),
then
Art(a)=∏Frob_(P_i)^(n_i).
This converts ideal factorisation into Galois action.
Computationally, once prime-ideal Artin symbols are known, arbitrary coprime ideals are handled by linear combination in the ray class group.
4. The rational-field prototype
Kronecker–Weber says every finite abelian extension of Q lies inside a cyclotomic field.
For
L=Q(ζ_m),
the Galois group is
Gal(L/Q) ≅ (Z/mZ)^×.
An integer a coprime to m acts by
σ_a(ζ_m)=ζ_m^a.
5. Worked example: Q(ζ5)
Let
L=Q(ζ_5).
Then
Gal(L/Q) ≅ (Z/5Z)^× ≅ C4.
Guide41 computed
Cl_(5∞)(Q) ≅ (Z/5Z)^×.
The full ray class field for the modulus 5∞ is therefore Q(ζ5).
6. Frobenius at p=2
The rational prime2 is unramified in Q(ζ5).
Its Artin element corresponds to
2 mod5.
Hence
Frob_2(ζ5)=ζ5².
Since2 has order4 modulo5, the Frobenius element has order4 in the Galois group.
7. Frobenius at p=11
Now
11≡1 (mod5).
Therefore
Frob_11=1.
So11 splits completely in Q(ζ5).
This recovers a classical cyclotomic splitting rule from the Artin map.
8. Residue degree from Frobenius order
For an unramified prime in a finite Galois extension, the order of the Frobenius element equals the residue degree.
In Q(ζ5):
ord_5(2)=4
so2 has residue degree4, while
ord_5(11)=1
so11 splits completely.
The same multiplicative-order calculations from Guide29 now describe prime splitting in a field extension.
9. Subgroups give subfields
Let
G=(Z/5Z)^×={1,2,3,4}.
The subgroup
H={1,4}={±1}
has index2.
Its fixed field inside Q(ζ5) has degree2 over Q and is
Q(ζ5+ζ5^(-1))=Q(√5).
10. Finite modulus 5 revisited
Guide41 found
Cl_5(Q) ≅ (Z/5Z)^×/{±1} ≅ C2.
This quotient is exactly the Galois group of the real quadratic subfield Q(√5).
The difference between modulus 5 and modulus 5∞ is therefore visible geometrically: including the real place allows the full complex cyclotomic extension rather than only its real subfield.
11. Hilbert class field
The Hilbert class field H_K is the maximal unramified abelian extension of K.
Its Galois group is canonically isomorphic to the ordinary ideal class group:
Gal(H_K/K) ≅ Cl(K).
Thus the ordinary class group is already a class-field Galois group—the ray class groups generalise this by permitting controlled ramification.
12. Principal ideals in the kernel
Why does the Artin map factor through a ray class group?
Because principal ideals generated by elements satisfying the ray congruence/sign conditions act trivially in the corresponding class field.
This is the reciprocity law at the heart of the construction: the correct principal subgroup is precisely the kernel required to turn ideals into a finite Galois group.
13. Conductors and ramification
If P does not divide the conductor, it is unramified and has a Frobenius element.
If P divides the conductor, ramification prevents an ordinary Frobenius element from being defined in the same way.
The conductor therefore records exactly where the Artin map must exclude prime ideals from its ideal domain.
14. Reciprocity as a quotient map
Computationally, class field theory becomes manageable because
Cl_m(K)
is a finite abelian group with an explicit presentation.
Choose a subgroup H. Then the quotient
Cl_m(K)/H
gives the desired abstract Galois group.
The harder step is constructing a defining polynomial for the corresponding field.
15. Constructing class fields
Major computational methods include:
complex multiplication for imaginary quadratic fields Kummer theory when enough roots of unity are present analytic class invariants Stark-unit methods in suitable cases generic ray class field routines combining local and global data.
No single construction method is uniformly best in every number field.
16. Kummer theory
If K contains the n-th roots of unity, cyclic extensions of degree dividing n can often be written
L=K(α^(1/n)).
Class-field conditions then determine which α produce the desired ramification and Artin behaviour.
This converts an abstract subgroup problem into explicit radical construction where the hypotheses permit it.
17. Imaginary quadratic class fields
For imaginary quadratic K, complex multiplication gives especially explicit class fields.
Hilbert class polynomials and more general ring/ray class invariants produce algebraic numbers whose fields realise class groups.
This connects Guide36’s CM curve construction back to class field theory.
18. Frobenius verification
Suppose a candidate field L has been constructed.
To verify the predicted Artin map, choose several unramified prime ideals P.
Factor P in L and compare the residue degree with the predicted order of Art(P) in the quotient group.
For completely split predicted classes, verify that every residue degree is one.
19. Chebotarev as a statistical check
In an abelian extension, primes are distributed among Frobenius elements according to Chebotarev density.
Each element of the Galois group has natural density 1/|G| among unramified primes.
Finite samples are not proofs, but systematic deviation in a large computational sample can reveal a broken reciprocity map or incorrect field model.
20. Reverse problem: identify an abelian extension
Given L/K:
verify L/K is finite Galois and abelian compute ramified primes bound / compute the conductor compute a suitable ray class group compute Artin symbols of test primes determine the kernel subgroup H verify [Cl_m(K):H]=[L:K].
This reconstructs the extension as a class field.
21. Local conductor information
Ramification groups and local unit filtrations control conductor exponents at each finite prime.
A global conductor is assembled from these local exponents plus real-place data.
This is why local field arithmetic from Guide39 is part of an explicit class-field implementation.
22. Artin map coordinates
If the ray class group has invariant factors
C_(n1)×...×C_(nr),
represent an ideal by its coordinate vector in this group.
Then a quotient by H is ordinary finite abelian-group linear algebra.
Frobenius computation becomes coordinate reduction followed by the chosen group-to-automorphism dictionary.
23. Exact sequence viewpoint
Computational class field theory is layered:
units and residue units → ray class group → subgroup quotient → abstract Galois group → explicit field → verified Artin action.
Each arrow has its own data structure and proof obligation.
24. Why class field theory stops at abelian extensions
The global reciprocity map is abelian by construction: it passes through an abelian ideal class group.
Nonabelian extensions require richer structures. Their splitting behaviour can still be studied with Frobenius conjugacy classes, but they are not classified by ray class groups.
25. Computation versus existence
Class field theory proves that a class field corresponding to a subgroup exists and is unique up to K-isomorphism.
An explicit algorithm must still produce a polynomial, embeddings or generators representing that field.
The existence theorem and the construction algorithm are therefore distinct stages.
26. Certification
A strong computational receipt for an abelian class field includes:
base field K modulus m ray class group presentation subgroup H quotient G=Cl_m/H candidate defining polynomial for L degree check Galois / abelian verification ramification check Artin/Frobenius tests at several primes conductor verification.
27. Common mistakes
1. Treating every Galois extension as a class field. 2. Ignoring the conductor. 3. Defining Frobenius at ramified primes as though nothing changes. 4. Confusing a ray class group with the Galois group before quotienting by H.
5. Assuming an abstract quotient automatically supplies a defining polynomial. 6. Omitting infinite-place conditions. 7. Using prime splitting alone as a proof without checking enough structural data. 8. Forgetting that Artin symbols multiply through ideal factorisation.
28. Practice set
1. State the Artin reciprocity quotient. 2. Define Frobenius at an unramified prime. 3. What is Gal(Q(ζ5)/Q)? 4. What does Frob_2 do to ζ5?
5. Why does11 split completely in Q(ζ5)? 6. Which subgroup fixes Q(√5)? 7. What is the Hilbert class field? 8. What is the relation between its Galois group and Cl(K)?
9. What does the conductor control? 10. Name one explicit class-field construction method. 11. How can Frobenius predictions be tested? 12. Why are ray class groups insufficient for nonabelian classification?
29. Answers
1. Gal(L/K)≅Cl_m(K)/H. 2. The automorphism acting on the residue field by x↦x^(N(P)). 3. (Z/5Z)^×≅C4. 4. ζ5↦ζ5².
5. 11≡1 mod5, so its Frobenius is identity. 6. {±1}. 7. The maximal unramified abelian extension of K. 8. Gal(H_K/K)≅Cl(K).
9. The smallest modulus through which reciprocity factors and the allowed ramification. 10. CM, Kummer theory or class-invariant methods. 11. Factor unramified primes in the candidate field and compare residue degrees with Artin orders. 12. Ray class groups are abelian.
Sources and further study
Henri Cohen, A Survey of Computational Class Field Theory, describes ray class groups as the first computational layer for explicit abelian extensions and surveys field-construction methods. Classical global class field theory provides the existence and reciprocity theorems underlying these algorithms.
Continue through Batch 11
Return to Guide 41. Continue to Guide 43: p-Adic Logarithms and Guide 44: PSLQ.