The Four Core Computational Tasks of Number Fields
The four fundamental computational problems for a number field, their dependencies, and what counts as a complete answer to each.
Engineering / MathematicsOrientation9 min readKV-MATH-0504
Given a number field presented as the quotient of the rationals by an irreducible polynomial, four computational tasks account for nearly everything one wants to know. They are strictly ordered by dependency: each needs the ones before it.
The four tasks
The four core tasks, in dependency order
Compute the maximal orderFind an integral basis for the ring of integers. Everything else is expressed relative to this basis, so an error here invalidates all downstream work. See the Round 2 algorithm.
Decompose primesFor a rational prime p, find the prime ideals above it with their ramification indices and residue degrees. See the simple case and Buchmann-Lenstra.
Compute the class groupDetermine the group structure as a product of cyclic groups, with explicit ideal generators. See relation matrices.
Compute units and the regulatorFind a system of fundamental units and the regulator. In practice this falls out of the same relation collection as the class group. See regulator recovery.
Why the ordering is strict
The dependency is not stylistic. Each task consumes the output of its predecessor in an essential way.
Dependencies between the four tasks
Task
Consumes
Why it cannot be reordered
Prime decomposition
Maximal order
Prime ideals are ideals of the maximal order. Decomposing in a non-maximal order gives a different and generally wrong answer.
Class group
Prime decomposition
The factor base consists of prime ideals of small norm. You cannot assemble it without decomposition.
Units and regulator
Class group relations
Relations that are trivial in the class group are exactly the ones that yield units. Both come from the same matrix.
What counts as a complete answer
Maximal order
An integral basis expressed in terms of a power basis, plus the field discriminant. A basis alone is not enough — the discriminant is what lets you verify maximality.
Prime decomposition
For each prime above p: a two-element or HNF representation of the ideal, its ramification index e, and its residue degree f, satisfying the degree relation.
Class group
The invariant factor decomposition, plus an explicit ideal generating each cyclic factor. The order alone (the class number) is a weaker result.
Units
A system of fundamental units in the standard representation, plus the regulator to sufficient precision, plus the roots of unity.
Conditionality
The first two tasks are unconditional and deterministic. The last two are usually computed under the Generalised Riemann Hypothesis, because unconditional bounds on the factor base are impractically large. This is a real distinction that must be recorded with the result.
Conditionality of the four tasks
Task
Status
Note
Maximal order
Unconditional
Requires factoring the polynomial discriminant, which may be hard but is not conditional
Prime decomposition
Unconditional
Deterministic
Class group
Conditional on GRH in practice
Unconditional verification is possible but expensive
Regulator
Conditional on GRH in practice
Can be checked against the analytic class number formula
Verification
Each task admits an independent check, and using them is strongly advised because the failure modes are quiet.
Maximal order
Verify that the index-squared divides the polynomial discriminant and that the quotient is the field discriminant.
Prime decomposition
Check that the sum of e times f over all primes above p equals the field degree. This catches most errors immediately.
Class group and regulator
Compare the product of class number and regulator against the analytic class number formula. See verification.
Frequently Asked Questions
Can the class group be computed without the maximal order?
Not meaningfully. The class group is defined in terms of fractional ideals of the maximal order. Ring class groups of non-maximal orders are a different and coarser invariant.
Is the class number enough, or do I need the group structure?
It depends on the application, but the structure is strictly more informative and usually costs no extra — it falls out of the Smith normal form of the relation matrix that you had to compute anyway.
How expensive is each task in practice?
For fields of small degree and moderate discriminant, the first two are fast. The class group and regulator dominate, and their cost grows sub-exponentially in the discriminant, so they set the practical limit on field size.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.9.3. 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.
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 The Four Core Computational Tasks of Number Fields. 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 The Four Core Computational Tasks of Number Fields as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—four, computational, tasks, number, counts—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 The Four Core Computational Tasks of Number Fields?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which assumption about four would change the result most if it proved false?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What evidence would allow an independent reviewer to reproduce or challenge the conclusion?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which boundary, exception or failure case has not yet been tested?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What must be handed over, monitored or reviewed after the immediate work is complete?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Authoritative references and use notes
The sources below were selected as institutional or primary guidance for the broader practice. They support the handbook method; they do not imply that every statement or clause in a source applies to every project. Confirm the current edition, jurisdiction, contract and application before treating any requirement as mandatory.
MIT OpenCourseWare — Number Theory I — Massachusetts Institute of Technology. Used for algebraic and analytic number theory. Accessed 2026-08-13.
MIT OpenCourseWare — Algebra I — Massachusetts Institute of Technology. Used for groups, vector spaces, linear transformations and linear groups. Accessed 2026-08-13.
