Class Groups, Units and Regulators
The Ideal Class Group
The class group as the obstruction to unique factorisation, its finiteness, and what a complete computation must produce.
Engineering / MathematicsClass Groups, Units and Regulators2 min readKV-MATH-0593
The class group measures the failure of unique factorisation of elements in the maximal order. It is finite, and computing it is one of the two hard problems of the subject.
Definition
- Trivial class group
- Class number one. Every ideal is principal, and elements factor uniquely up to units.
- Ideal class
- Two ideals are in the same class when their quotient is principal.
- Class number
- The order of the group. A weaker invariant than the structure.
Finiteness
Every ideal class contains an ideal of norm below the Minkowski bound. Since there are finitely many ideals below any norm bound, the class group is finite.
What a complete computation produces
| Output | Why it is needed |
|---|---|
| Invariant factor decomposition | The group structure, not merely its order |
| An ideal generating each cyclic factor | Without generators the structure cannot be used |
| The order of each generator | Follows from the invariant factors |
| Conditionality flag | Whether the result assumes GRH |
Behaviour
Class numbers of imaginary quadratic fields grow roughly like the square root of the discriminant. Real quadratic fields behave quite differently: class numbers are frequently very small while the regulator is large, and the product of the two is what the analytic formula controls.
Computation
For quadratic fields, classical methods via binary quadratic forms are available. In general, the sub-exponential relation method applies — see Buchmann's algorithm.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.9.1. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.