Computing the Maximal Order: the Round 2 Algorithm

Executive summary

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.

The factoring obstruction

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.

AlgorithmComputing the p-radicalin: order O, prime p → out: the p-radical I_p

Let q be the smallest power of p with q ≥ n, where n is the degree.
Compute the matrix of the map x ↦ x^q on O/pO, an F_p-linear map. Linear because the Frobenius is additive in characteristic p.
The p-radical I_p is the preimage in O of the kernel of that map.
Return I_p as an HNF module containing pO.

The ring of multipliers of the radical is then the candidate larger order:

O′ = { x ∈ K : x I_p ⊆ I_p }

Why the ring of multipliers is larger

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

AlgorithmRound 2 (p-maximal overorder)in: order O, prime p → out: the p-maximal overorder of O

Apply the Dedekind criterion. If O is p-maximal, return O. Cheap test first — it succeeds for most primes.
Compute the p-radical I_p.
Compute the ring of multipliers O′ of I_p, 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 4 and the alternatives

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

How to record a maximal order computation
Situation	Correct claim
disc(T) fully factored	ℤ_K determined unconditionally
Unfactored part proved squarefree	ℤ_K 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

Downstream consequences of a non-maximal order

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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ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Computing the Maximal Order: the Round 2 Algorithm. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat Computing the Maximal Order: the Round 2 Algorithm as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—maximal, round, algorithm, section, ring—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying Computing the Maximal Order: the Round 2 Algorithm?