Sub-exponential Class Group Computation
Regulator and Fundamental Unit Recovery
Extracting fundamental units and the regulator from the kernel of the relation matrix, and confirming the unit system is fundamental.
Engineering / MathematicsSub-exponential Class Group Computation2 min readKV-MATH-0630
Units fall out of the same relation matrix that gives the class group. A relation that is trivial as an ideal identity means the generator involved is a unit, and the collection of such generators spans the unit lattice.
Where units come from
Each relation records a product of ideals equal to a principal ideal with a known generator. A combination of relations whose exponent vectors cancel gives a product of generators generating the trivial ideal — that is, a unit.
The procedure
Recovering units and the regulator
- Compute the kernelOf the relation matrix over the integers.
- Form the unitsEach kernel vector gives a product of recorded generators.
- Embed logarithmicallyUsing the accumulated real vectors — see the logarithmic embedding.
- Reduce the unit latticeApply LLL to obtain a smaller and more nearly fundamental system.
- Compute the regulatorAs the determinant of the reduced logarithmic matrix with one column deleted.
- Check fundamentalityAgainst the analytic class number formula.
Units are never written out
The index problem
A system obtained this way is independent but may not be fundamental. The regulator computed is then an integer multiple of the true one, and determining that integer is the final step.
| Approach | Effect |
|---|---|
| Lattice reduction on the unit lattice | Finds smaller units; reduces but may not eliminate the index |
| More relations | Often produces the missing units directly |
| Comparison with the analytic formula | Reveals the index as a ratio |
| Bounded search under GRH | Makes an exhaustive search finite |
Precision
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 6.5.3. 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.