Maximal at every prime implies maximal
An order is the maximal order exactly when it is maximal at every prime dividing the index, and only primes whose square divides the polynomial discriminant can do so. Round 2 therefore factors that discriminant, tests each candidate prime with the Dedekind criterion, and for each failure computes the p-radical and its ring of multipliers — a strictly larger order. Iterating at each prime terminates because the index drops by a factor of p each time.
Learning objectives
- State the Pohst–Zassenhaus theorem and its role.
- Apply the Dedekind criterion as a cheap maximality test.
- Compute the p-radical and the ring of multipliers.
- Trace the Round 2 loop and prove its termination.
- Report an order correctly when the discriminant is not fully factored.
Section 01The Pohst–Zassenhaus theorem
An order O is maximal if and only if it is p-maximal for every prime p with p2 dividing disc(O). This localises the problem: instead of one global search, a finite list of independent local computations.
- Stage 01Compute disc(T)By the subresultant algorithm.
- Stage 02Factor the square partOnly primes whose square divides the discriminant can divide the index — the squarefree part is discarded immediately.
- Stage 03Test each candidateApply the Dedekind criterion. Most primes pass and need no further work.
- Stage 04Enlarge where neededFor each failing prime, compute the p-maximal overorder; merge all results into a single integral basis by Hermite normal form.
Step 2 is the bottleneck, and it is not an implementation weakness — no algorithm is known that determines the maximal order without it. Where the discriminant cannot be fully factored, the result must be reported as maximal only at the primes that were resolved.
Section 02The p-radical and the ring of multipliers
The p-radical of O is the set of elements some power of which lies in pO — equivalently, the nilradical of O/pO pulled back to O.
- Let q be the smallest power of p with q ≥ n, where n is the degree.
- Compute the matrix of the map x ↦ xq on O/pO, an Fp-linear map. Linear because the Frobenius is additive in characteristic p.
- The p-radical Ip is the preimage in O of the kernel of that map.
- Return Ip as an HNF module containing pO.
The ring of multipliers of the radical is then the candidate larger order:
If O is not p-maximal, the radical fails to be invertible, and elements outside O multiply it into itself. Those elements are algebraic integers not in O, so O′ strictly contains O. When O is already p-maximal the construction returns O itself, which is the loop's termination signal.
Section 03The Round 2 algorithm
- Apply the Dedekind criterion. If O is p-maximal, return O. Cheap test first — it succeeds for most primes.
- Compute the p-radical Ip.
- Compute the ring of multipliers O′ of Ip, as the kernel of a linear map expressed via the regular representation.
- If O′ = O, then O is p-maximal; return O.
- Otherwise set O ← O′ and return to step 2. The index [O′ : O] is a positive power of p, so disc drops by at least p2.
Round 2 is the standard because it is simple and robust. The Round 4 algorithm of Ford and Letard uses Newton polygons and Montes-style techniques to obtain the same result faster for high-degree fields and for primes with large ramification, at the cost of considerably more intricate implementation.
Section 04Reporting the result
| Situation | Correct claim |
|---|---|
| disc(T) fully factored | ℤK determined unconditionally |
| Unfactored part proved squarefree | ℤK determined unconditionally |
| Unfactored part remains | Order maximal at all primes below the factoring bound; field discriminant known only up to a square factor |
| Only trial division performed | Maximal at small primes; treat the result as provisional |
Every later computation — prime decomposition, class group, units — assumes maximality. Running them on a non-maximal order gives well-formed and incorrect answers. The conditionality must therefore propagate with the data, not be recorded once and forgotten.
ReferenceFrequently asked questions
How often does the Dedekind criterion succeed immediately?
For most primes. It fails only when the factorisation of the defining polynomial modulo p has repeated factors and an additional gcd condition holds, which is uncommon — so the expensive radical computation typically runs for only a handful of primes.
Can the maximal order be computed for large degree fields?
Yes, subject to the discriminant factoring. The linear algebra is polynomial in the degree; the practical limit is almost always the size of the discriminant rather than the degree itself.
What if the defining polynomial has huge coefficients?
Reduce it first. Polynomial reduction — using LLL on the lattice of the order to find a generator of small height — can shrink the discriminant enormously and may convert an infeasible factorisation into an easy one.
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