Maximal Orders and Decomposition II
The Buchmann-Lenstra Prime Decomposition Method
Decomposing any prime, including those dividing the index, by splitting the algebra of the order modulo that prime.
Engineering / MathematicsMaximal Orders and Decomposition II2 min readKV-MATH-0617
The Buchmann-Lenstra method decomposes a prime without any assumption about the index. It works with the order modulo the prime as an algebra and splits it, which handles the cases where polynomial factorisation is insufficient.
Why the simple method fails
The method
Buchmann-Lenstra prime decomposition
- Reduce the orderRepresent the maximal order modulo p as an algebra over the field with p elements.
- Compute the radicalThe nilpotent elements — see radical computation.
- QuotientThe quotient by the radical is a product of finite fields, one per prime above p.
- SplitFind the idempotents — see algebra splitting.
- LiftPull each component back to a prime ideal of the order.
- Compute ramificationDetermine each ramification index by valuation — see valuations.
Reading the invariants
| Invariant | Where it comes from |
|---|---|
| Number of primes | Number of simple components of the quotient |
| Residue degree | Degree of each component over the prime field |
| Ramification index | Computed by valuation after the primes are constructed |
| Verification | Sum of e times f equals the field degree |
Cost
Choosing between the methods
- Run the Dedekind criterion at p
- Passes — use simple decompositionNearly free; the factorisation is already computed
- Fails — p divides the index
- Moderate degree — Buchmann-Lenstra
- High degree or heavy ramification — Newton polygons
Correctness
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 6.2.2-6.2.5. 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.