Maximal Orders and Decomposition II
The Dedekind Criterion for p-Maximality
A cheap modular test deciding whether an order is maximal at a given prime, without computing the maximal order.
Engineering / MathematicsMaximal Orders and Decomposition II2 min readKV-MATH-0612
The Dedekind criterion decides p-maximality of the equation order using only polynomial arithmetic modulo p. It is cheap, exact, and avoids the enlargement machinery in the overwhelming majority of cases.
The test
The Dedekind criterion
- Factor modulo pFactor the defining polynomial modulo p into irreducible factors with multiplicities.
- Form the radical productTake the product of the distinct irreducible factors.
- LiftLift both the factorisation and the radical product to the integers.
- Form the test quantityCompute the difference between the polynomial and the product of lifts, divided by p.
- Take a GCDThe order is p-maximal exactly when a certain GCD modulo p is trivial.
What it decides
| Outcome | Meaning | Next step |
|---|---|---|
| GCD trivial | The equation order is maximal at p | No work needed; use simple decomposition |
| GCD non-trivial | p divides the index | Enlarge via Round 2 |
Why it matters practically
It also decides decomposition
When the criterion passes, the factorisation computed during the test is exactly what simple decomposition requires, so the prime's decomposition comes out at no additional cost.
Relation to ramification
A repeated factor modulo p signals ramification but not necessarily non-maximality. The two are distinct: a prime can be ramified and the order still maximal there. Conflating them causes unnecessary enlargement work.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 6.1.2. 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.