Quadratic Fields and Binary Quadratic Forms

Q: Why is the discriminant sometimes 4d?

Because when d is congruent to 2 or 3 modulo 4, the element (1 + √d)/2 is not an algebraic integer, so the integral basis is 1, √d and the discriminant of that basis is 4d. The congruence condition determines which basis is correct.

Q: What is a fundamental discriminant?

An integer congruent to 1 modulo 4 and squarefree, or four times a squarefree integer congruent to 2 or 3 modulo 4. These are exactly the discriminants of quadratic fields; other discriminants belong to non-maximal orders.

Q: Should I use forms or ideals in an implementation?

For quadratic fields specifically, forms — the arithmetic is smaller, faster and entirely integral. For any code intended to generalise to higher degree, ideals, since the form language does not extend.

Executive summary

Two languages for one group

A quadratic field is ℚ(√d) for squarefree d, with discriminant D equal to d or 4d according to a congruence. Prime decomposition is decided by a single Kronecker symbol. Ideal classes correspond exactly to equivalence classes of binary quadratic forms of discriminant D, with ideal multiplication corresponding to Gauss composition — and the form language, being purely integral, is what practical algorithms actually manipulate.

Learning objectives

Compute the discriminant and integral basis of a quadratic field.
Decide the decomposition of a prime by the Kronecker symbol.
Reduce a binary quadratic form and count reduced forms.
Explain the correspondence between form classes and ideal classes.
Distinguish fundamental from non-fundamental discriminants.

Section 01Quadratic fields

For squarefree d ≠ 0, 1, the field K = ℚ(√d) has

Discriminant and integral basis
Condition	Discriminant D	Integral basis
d ≡ 1 (mod 4)	d	1, (1 + √d)/2
d ≡ 2 or 3 (mod 4)	4d	1, √d

Discriminants arising this way are the fundamental discriminants. A general discriminant D = f²D₀ corresponds to the non-maximal order of conductor f. The field is imaginary when D < 0 and real when D > 0, and that sign governs everything: unit rank 0 versus 1, finite versus infinite unit group, and a class number computation that is straightforward versus one that is entangled with the regulator.

Prime decomposition in one symbol

For p not dividing D, the Kronecker symbol (D/p) decides: +1 means p splits into two primes, −1 means p is inert, and 0 means p ramifies. No polynomial factorisation is needed — which is why quadratic fields are the natural testbed for every algorithm in the subject.

Section 02Binary quadratic forms

A binary quadratic form is f(x, y) = ax² + bxy + cy², written (a, b, c), with discriminant D = b² − 4ac. Two forms are equivalent when related by a unimodular change of variables.

AlgorithmReduction of a positive definite form (D < 0)in: a positive definite form of discriminant D → out: the reduced form

Normalise: replace b by its representative in (−a, a] by translating x.
If a > c, swap: set (a, b, c) ← (c, −b, a). This is the inversion x ↦ −y, y ↦ x.
Repeat from step 1 until |b| ≤ a ≤ c.
If a = c or |b| = a, fix the sign convention by requiring b ≥ 0.
Return the reduced form — unique in its equivalence class.

Indefinite forms behave differently

For D > 0 reduction does not produce a unique representative. Instead the reduced forms in a class form a cycle, traversed by the continued fraction expansion of a quadratic irrational. The length of that cycle is tied to the regulator — which is why real quadratic fields are computationally harder.

Section 03The correspondence with ideals

FormsClassical language

Forms (a, b, c) of discriminant D, up to unimodular equivalence, with Gauss composition as the group law. Entirely integral, so arithmetic is exact and fast.

IdealsModern language

Ideals of the quadratic order of discriminant D, up to principal ideals, with ideal multiplication as the group law. Generalises to higher degree.

The correspondence sends the form (a, b, c) to the ideal with ℤ-basis a and (−b + √D)/2. It is a group isomorphism: composition of forms corresponds to multiplication of ideals, reduction of forms to reduction of ideals, and the class number of forms to the class number of the order.

Two conventions for the class group of forms

Some sources use proper (unimodular, determinant +1) equivalence and some allow determinant −1. For D < 0 the two differ, since the improper equivalence identifies a class with its inverse. Class number tables must be read with the convention in mind, or genus-level discrepancies will appear.

Section 04Composition

Gauss composition is the group law in the form language. The modern formulation uses the united-forms construction: given two forms with the same discriminant, solve a small system of congruences by the extended Euclidean algorithm to produce the composed form, then reduce.

O(log D)cost of one composition

a ≤ √(|D|/3)size bound after reduction

Exact integersno floating point required

Why this matters for algorithms

Composition plus reduction gives a group operation on objects of bounded size at a cost comparable to a GCD. That is what makes baby-step giant-step, Shanks's class group method and the sub-exponential algorithms practical for quadratic fields long before comparable methods existed in higher degree.

ReferenceFrequently asked questions

Why is the discriminant sometimes 4d?

Because when d is congruent to 2 or 3 modulo 4, the element (1 + √d)/2 is not an algebraic integer, so the integral basis is 1, √d and the discriminant of that basis is 4d. The congruence condition determines which basis is correct.

What is a fundamental discriminant?

An integer congruent to 1 modulo 4 and squarefree, or four times a squarefree integer congruent to 2 or 3 modulo 4. These are exactly the discriminants of quadratic fields; other discriminants belong to non-maximal orders.

Should I use forms or ideals in an implementation?

For quadratic fields specifically, forms — the arithmetic is smaller, faster and entirely integral. For any code intended to generalise to higher degree, ideals, since the form language does not extend.

NavigateContinue in this stream

Curated next steps from this page. The site also surfaces algorithmically related reading below.

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 Quadratic Fields and Binary Quadratic Forms. 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 Quadratic Fields and Binary Quadratic Forms as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—quadratic, fields, binary, forms, section—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 Quadratic Fields and Binary Quadratic Forms?