Class Group and Unit Computation in General Number Fields

Executive summary

One relation matrix, four answers

The general algorithm chooses a factor base of small prime ideals, generates random ideals that factor completely over it, and records each factorisation as a relation. The Smith normal form of the resulting matrix gives the class group; its kernel gives the units and hence the regulator. Under GRH the factor base can be taken far smaller than Minkowski's bound, which is what makes the method sub-exponential — and what makes the result conditional.

Learning objectives

Select a factor base and state the GRH-conditional bound.
Generate relations by reducing random ideals.
Extract the class group from the relation matrix.
Extract units and the regulator from its kernel.
Verify the result and solve the principal ideal problem.

Section 01Factor base selection

The factor base is the set of prime ideals of norm below a bound B. Unconditionally B must reach Minkowski's bound to guarantee generation of the class group; under GRH a bound of order (log|d_K|)² suffices.

Minkowskiunconditional, exponentially large

O(log²|d_K|)GRH-conditional, practical

L[1/2]resulting complexity

The bound decides the conditionality

Choosing the GRH bound is what makes the algorithm feasible and what makes its output conditional. A computation run to Minkowski's bound is unconditional but usually infeasible; one run to the GRH bound is fast and must be reported as conditional. There is no middle option, and the choice must be recorded.

Section 02Relation collection

AlgorithmRelation collectionin: factor base → out: a relation matrix and generating elements

Choose random exponents and form the ideal 𝔼 = ∏ 𝔽_i^a_i over the factor base.
Reduce 𝔼: find a short element α in the ideal lattice by LLL and replace 𝔼 by (α)𝔼⁻¹. Reduction keeps norms bounded by roughly √|d_K|.
Attempt to factor N(𝔼) over the factor base primes. If it does not split completely, discard and return to step 1.
Record the relation: the exponent vector, together with the element α and its logarithmic embedding.
Repeat until there are comfortably more relations than factor base elements.

Reduction is what makes this possible

Without ideal reduction the norms grow with each multiplication and never factor over a fixed factor base. LLL-based reduction bounds them, which is the precise point at which lattice reduction becomes indispensable to class group computation.

Section 03Extracting the invariants

From the matrixThe class group

The Smith normal form of the relation matrix gives the invariant factors, hence the structure of Cl(K) and the class number as their product.

From the kernelThe units and the regulator

A kernel vector corresponds to a product of the generating elements that is a unit. The logarithmic embeddings of independent such units form a matrix whose determinant is the regulator.

Stage 01Assemble the matrixRows are relations, columns are factor base primes.
Stage 02Smith normal formInvariant factors give the class group structure.
Stage 03KernelKernel vectors give units; take r independent ones.
Stage 04RegulatorDeterminant of the logarithmic embedding matrix, to certified precision.
Stage 05VerifyCompare hR against the analytic estimate; if short by a factor, collect more relations.

A subgroup is the default failure mode

Too few relations yields a proper subgroup of the class group and a proper subgroup of the units — and therefore a class number too small and a regulator too large, in a compensating way that leaves the product looking plausible. Only comparison against an independent analytic estimate reliably detects it.

Section 04The principal ideal problem

Deciding whether a given ideal is principal, and if so finding a generator, is a distinct problem solved with the same machinery: reduce the ideal, factor it over the factor base, and solve the resulting linear system against the relation matrix. A solution yields a generator as a product of the recorded elements.

Generators can be astronomically large

The generator produced is a product of powers of relation elements and may have enormous height. It is stored in compact form — as the exponent vector over the relation elements — and expanded only if genuinely required, which it usually is not.

ReferenceFrequently asked questions

How large must the factor base be?

Large enough that random reduced ideals factor over it with workable probability, and small enough that the linear algebra remains tractable. The optimum balances the two and is found by the same L-notation analysis that governs the quadratic sieve.

Why are both the class group and the units obtained together?

Because a relation says that a specific product of factor base ideals is principal, generated by a specific element. The exponent vectors give the class group; the elements attached to kernel combinations give units. The two are different readings of the same data.

Can the result ever be certified unconditionally?

Yes, by re-verifying with a factor base extended to Minkowski's bound, which is feasible only for small discriminants. Otherwise the standard practice is to report the result as correct under GRH, together with the analytic consistency check.

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ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Class Group and Unit Computation in General Number Fields. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat Class Group and Unit Computation in General Number Fields as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—relation, class, number, section, computation—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying Class Group and Unit Computation in General Number Fields?