Maximal Orders and Decomposition II
The Maximal Order Problem
The maximal order problem, its reduction to a prime-by-prime question, and the factoring obstruction that limits it.
Engineering / MathematicsMaximal Orders and Decomposition II2 min readKV-MATH-0610
Computing the maximal order is the first of the four core tasks and the prerequisite for every other. The problem reduces cleanly to a question about individual primes, but the reduction itself requires a factorisation.
The problem
Given a defining polynomial, find an integral basis for the ring of all algebraic integers in the field.
Reduction to primes
An order is maximal exactly when it is maximal at every prime. Since the index appears squared in the discriminant relation, only primes whose square divides the polynomial discriminant can be involved.
The maximal order computation
- Compute the polynomial discriminantBy resultant with the derivative.
- Find the square divisorsRequires factoring, or at least finding the square part.
- Test p-maximalityFor each candidate prime — see the Dedekind criterion.
- Enlarge where neededBy Round 2.
- VerifyThe index squared must divide the polynomial discriminant exactly as expected.
The factoring obstruction
Partial maximality
If the discriminant cannot be fully factored, an order maximal at all known primes can still be produced, with the unfactored cofactor recorded. Any computation involving only the known primes remains valid.
| Situation | What remains valid |
|---|---|
| Fully factored discriminant | Everything |
| Square-free cofactor proved | Everything — a squarefree cofactor cannot divide the index |
| Unfactored cofactor | Prime decomposition at known primes; class group results are unsafe |
The single best mitigation
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 6.1. 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.