Discriminants and Integral Bases

Executive summary

The gap between ℤ[θ] and ℤK is measured by a square

The obvious ring inside a number field is ℤ[θ], generated by the defining root. It is rarely the full ring of integers. The two differ by a finite index whose square relates the polynomial discriminant to the field discriminant, so identifying ℤ_K means finding the square factors of disc(T) — a factoring problem, and the reason this step dominates the cost of setting up a number field.

Learning objectives

Define the discriminant of a basis and of a field.
Relate the polynomial discriminant to the field discriminant via the index.
Explain why the maximal order computation is as hard as factoring.
Apply the Dedekind criterion to test maximality at a prime.
Interpret the discriminant as the ramification locus.

Section 01Discriminant of a basis

For a ℚ-basis (ω₁, …, ω_n) of K, the discriminant is

disc(ω₁, …, ω_n) = det(Tr(ω_iω_j)) = det(σ_i(ω_j))²

Changing basis by a matrix U multiplies the discriminant by (det U)². For two integral bases U is unimodular, so the discriminant is the same — it is an invariant of the field, written d_K.

d_Kfield discriminant — a basis-independent invariant

signnegative exactly when r<sub>2</sub> is odd

≡ 0 or 1modulo 4, by Stickelberger's theorem

Section 02The index and the factoring obstruction

The equation order ℤ[θ] sits inside ℤ_K with finite index, and

disc(T) = [ℤ_K : ℤ[θ]]² · d_K

So the polynomial discriminant — easy to compute by a resultant — equals the field discriminant times a perfect square. Extracting d_K requires knowing which square divisors of disc(T) belong to the index.

This is where the difficulty lives

Deciding whether a large square divides the index requires knowing the factorisation of disc(T), which can be an enormous integer even for modest defining polynomials. No algorithm is known that determines ℤ_K unconditionally without this factorisation. In practice systems factor as far as is feasible and return an order that is maximal at all primes that were resolved — and this qualification must be recorded with the result.

Stage 01Compute disc(T)By the subresultant algorithm — exact and fast.
Stage 02Factor the square partTrial division, then Pollard ρ, ECM and the quadratic sieve as needed. The bottleneck.
Stage 03Test each primeFor each p whose square divides disc(T), decide whether ℤ[θ] is maximal at p.
Stage 04Enlarge where neededWhere it is not, compute the p-maximal overorder and merge into a single integral basis.

Section 03The Dedekind criterion

Before doing expensive work at a prime p, a cheap test decides whether ℤ[θ] is already maximal there. The criterion works entirely with polynomial factorisation modulo p.

AlgorithmDedekind criterion at pin: T, prime p → out: whether ℤ[θ] is p-maximal

Factor T modulo p as ∏ g̅_i^e_i.
If every e_i = 1, then p does not divide the index — ℤ[θ] is maximal at p. Stop.
Otherwise set g ← ∏ g_i and h ← T/g computed modulo p, using arbitrary lifts.
Set F ← (gh − T)/p, reduced modulo p.
ℤ[θ] is maximal at p if and only if gcd(F̅, g̅, h̅) = 1 in F_p[x].

Only squares matter

A prime can divide the index only if its square divides disc(T). The squarefree part of the discriminant can therefore be ignored entirely, which usually removes most of the factoring burden before it begins.

Section 04Discriminant as ramification locus

A rational prime ramifies in K — meaning some prime ideal above it occurs with exponent greater than 1 — exactly when it divides d_K. Only finitely many primes ramify, and the discriminant lists them.

Consequence

Prime decomposition

For unramified p, factoring T modulo p directly gives the prime ideals above p — the simple algorithm applies.

Consequence

Minkowski's theorem

|d_K| > 1 for every field other than ℚ, so some prime always ramifies in a non-trivial extension.

Consequence

Field tabulation

Discriminant bounds constrain the fields of a given degree and signature, which is how number field tables are organised and searched.

ReferenceFrequently asked questions

Can the ring of integers be computed without factoring the discriminant?

Not unconditionally by any known method. There are algorithms that produce an order maximal at all primes below a chosen bound, which is often sufficient in practice, but the result must then be reported as conditional on the unfactored part being squarefree.

Is Z[theta] ever the full ring of integers?

Frequently — it happens exactly when the polynomial discriminant equals the field discriminant, that is when the index is 1. It is guaranteed for cyclotomic fields with the standard generator, and for quadratic fields when D is squarefree and congruent to 2 or 3 mod 4.

What does the Hermite–Minkowski theorem tell me?

That only finitely many number fields have discriminant below any given bound, and that the discriminant grows with the degree. This is what makes exhaustive tabulation of small-discriminant fields a finite and well-posed computation.

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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 Discriminants and Integral Bases. 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 Discriminants and Integral Bases as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—discriminant, section, integral, field, order—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 Discriminants and Integral Bases?