Class Groups, Units and the Regulator

Executive summary

h and R are computed together and verified together

The class group measures the failure of unique factorisation; its order is the class number h. The unit group is finitely generated of rank r₁ + r₂ − 1, and the covolume of its logarithmic lattice is the regulator R. Both come out of the same relation-collection computation, and the analytic class number formula links their product to a residue that can be estimated independently — giving the verification step that this subject otherwise lacks.

Learning objectives

Define the class group and interpret a trivial class number.
State Dirichlet's unit theorem and define the regulator.
Use Minkowski's bound to obtain a generating set of ideal classes.
Apply the analytic class number formula as a verification tool.
Report results correctly with respect to GRH.

Section 01The class group

The class group is the quotient of the fractional ideals by the principal ones:

Cl(K) = I(K) / P(K), h = |Cl(K)|

It is finite. Class number 1 means every ideal is principal, which is equivalent to unique factorisation of elements — so h is precisely the measure of how badly that property fails.

Minkowski's bound

Every ideal class contains an integral ideal of norm at most M_K = (4/π)^r₂(n!/nⁿ)√|d_K|. Hence the primes of norm below M_K generate the class group. This turns an infinite problem into a finite one — though for large discriminants the bound is far too big to use directly, which is why sub-exponential methods with GRH-conditional smaller bounds exist.

Section 02Units and the regulator

Dirichlet's unit theorem gives the structure of the unit group:

ℤ_K^× ≅ μ_K × ℤ^r, r = r₁ + r₂ − 1

Here μ_K is the finite group of roots of unity. The logarithmic embedding sends a unit to the vector of log|σ_i(ε)| weighted by 1 or 2; the image is a lattice of rank r, and the regulator is the absolute value of the determinant of any r×r minor of a matrix of fundamental units.

Unit rank by field type
Field	Signature	Rank r	Consequence
ℚ	(1, 0)	0	Units are ±1
Imaginary quadratic	(0, 1)	0	Finite unit group — regulator is 1 by convention
Real quadratic	(2, 0)	1	One fundamental unit; R = log ε
Complex cubic	(1, 1)	1	One fundamental unit
Totally real cubic	(3, 0)	2	Two fundamental units
Cyclotomic, p-th roots	(0, (p−1)/2)	(p−3)/2	Grows with p

The index-2 trap

A computed system of units may generate only a finite-index subgroup of the true unit group, in which case the regulator comes out as an integer multiple of the true value. This is the characteristic error mode, and the multiple is most often 2. It is detected only by the analytic check below.

Section 03The analytic class number formula

The Dedekind zeta function has a simple pole at s = 1 whose residue packages every invariant at once:

lim_s→1 (s−1)ζ_K(s) = 2^r₁(2π)^r₂ hR / (w √|d_K|)

The left side is estimated numerically from an Euler product over small primes; the right side contains the computed h and R. Agreement is strong evidence that both are correct; disagreement by a small integer factor is the signature of an incomplete relation set or a subgroup of units.

Stage 01Collect relationsFind principal ideals factoring over the factor base, recording exponent vectors and the generating elements.
Stage 02Extract hSmith normal form of the relation matrix gives the class group structure.
Stage 03Extract RThe kernel of the relation matrix yields units; their logarithmic embeddings give the regulator.
Stage 04VerifyCompare hR against the analytic estimate. If it is off by a factor k, continue collecting relations.

Report the GRH status explicitly

Sub-exponential algorithms use a factor base bounded by a GRH-conditional estimate, typically of order (log|d_K|)², far below Minkowski's bound. The results are correct under GRH; unconditional certification requires re-running with the Minkowski bound, which is usually infeasible. A computed class number should always carry its conditionality with it.

ReferenceFrequently asked questions

Why are h and R computed together?

Because a single relation-collection phase produces both: the relation matrix gives the class group, and its kernel gives the units. The analytic formula also constrains only the product, so verifying one requires the other.

What does class number 1 tell me?

That every ideal is principal and elements factor uniquely into irreducibles. It is a strong condition — only nine imaginary quadratic fields have it — and it makes many computations dramatically simpler.

Can the regulator be verified independently?

Partially. Lower bounds on the regulator exist in terms of the discriminant, and the analytic formula constrains the product hR. A regulator that is an exact small multiple of the analytically predicted value is the classic sign that a subgroup of units was found rather than the full group.

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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 Groups, Units and the Regulator. 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 Groups, Units and the Regulator as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—class, number, group, regulator, unit—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 Groups, Units and the Regulator?