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:
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.
Every ideal class contains an integral ideal of norm at most MK = (4/π)r2(n!/nn)√|dK|. Hence the primes of norm below MK 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:
Here μK is the finite group of roots of unity. The logarithmic embedding sends a unit to the vector of log|σi(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.
| 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 |
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:
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.
Sub-exponential algorithms use a factor base bounded by a GRH-conditional estimate, typically of order (log|dK|)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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