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GuidePublished 7 Aug 2026Updated 13 Aug 20268 min readBy Kevin Jogin
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Engineering  /  Mathematics  — Quadratic Residues

The Legendre Symbol

The Legendre symbol as a multiplicative character, its evaluation by Euler's criterion, and the supplementary laws.

Page KV-MATH-0409Reading time 3 minReviewed 2026-08-07Author Kevin Jogin

Executive summary

The Legendre symbol encodes quadratic residuosity modulo a prime as a value in plus one, minus one or zero. Its multiplicativity turns residuosity questions into arithmetic.

Together with the supplementary laws for minus one and two, it reduces most evaluations to a short computation.

Learning objectives

  1. Define the Legendre symbol and state its multiplicativity.
  2. Apply the supplementary laws.
  3. Evaluate symbols using Euler's criterion.

01Definition and multiplicativity

Definition

Legendre symbol

For an odd prime p and integer a:

(a|p) = 0 if p | a; = 1 if a is a non-zero quadratic residue mod p; = −1 otherwise.

Theorem

Multiplicativity

(ab|p) = (a|p)(b|p) for all integers a, b.

Multiplicativity follows directly from Euler's criterion, since the criterion expresses the symbol as a power and powers multiply. Structurally, the symbol is the homomorphism from Z_p* onto the two-element group whose kernel is the subgroup of residues.

The practical consequence is that a symbol can be evaluated by factoring the numerator and combining the symbols of the parts — though for large arguments the Jacobi symbol algorithm avoids the need to factor at all.

02The supplementary laws

Theorem

Supplementary laws

(−1|p) = 1 if p ≡ 1 (mod 4), and −1 if p ≡ 3 (mod 4).

(2|p) = 1 if p ≡ ±1 (mod 8), and −1 if p ≡ ±3 (mod 8).

Supplementary law values
p mod 8(−1|p)(2|p)(−2|p)
1+1+1+1
3−1−1+1
5+1−1−1
7−1+1−1
Note
The first supplementary law has a direct consequence used constantly: when p ≡ 3 (mod 4), minus one is a non-residue, so of any pair {x, −x} of square roots exactly one is itself a residue. This picks out a canonical root and is why such primes are preferred for square-root extraction.

03Evaluation

  1. Euler's criterionO(len(p)³)One modular exponentiation; simple but not fastest
  2. Factor and combineRequires factoring aImpractical for large arguments
  3. Jacobi symbol algorithmO(len(p)²)Euclid-like; the method actually used

Euler's criterion is the definition made computational, and it is correct but slower than necessary. The Jacobi algorithm proceeds like the Euclidean algorithm, using reciprocity to swap arguments and the supplementary laws to handle factors of two, and never factors anything.

Caution
Euler's criterion is valid only for prime moduli. Applying it to a composite gives a value that may be plus one for a non-residue, which is exactly the gap the quadratic residuosity assumption depends on.

04Frequently asked questions

Why is (0|p) defined as 0?

So that multiplicativity holds without exception. If it were left undefined, every identity would need a side condition excluding multiples of p.

Is the Legendre symbol a character?

Yes — it is the unique non-trivial real character on Z_p*, the quadratic character. This is the entry point to the theory of Dirichlet characters and L-functions.

Can the symbol be computed without knowing p is prime?

The Jacobi symbol generalises it to odd composite moduli and is computable without factoring. It agrees with the Legendre symbol when the modulus is prime but no longer determines residuosity.

Related pages

  • The Law of Quadratic Reciprocity
  • Quadratic Residues

Sources and method

Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 285-287.

This page carries the durable method layer only: definitions, constructions, algorithms, complexity results and selection criteria, authored originally for KEVOS. No text is transcribed or paraphrased from the source, and no numeric tables or benchmark data are reproduced — these are routed to live authoritative sources instead.

Author: Kevin Jogin. Last reviewed 2026-08-07.

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 Legendre Symbol. 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 Legendre Symbol as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—legendre, symbol, evaluation, supplementary, laws—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

  1. Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
  2. Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
  3. Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
  4. Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
  5. 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.

StageRecordQuality check
InputObjects, domain, notation, assumptionsEvery symbol is defined
MethodPermitted operation or cited result at each stepAll hypotheses hold
OutputExact result and representationCorrect type, domain and form
VerificationSubstitution, invariant or alternative derivationIndependent agreement
Boundary testZero, identity, degenerate or failed hypothesisScope 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 Legendre Symbol?

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 legendre 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 — Algebra I — Massachusetts Institute of Technology. Used for groups, vector spaces, linear transformations and linear groups. Accessed 2026-08-13.
  • The Stacks Project — table of contents — The Stacks Project. Used for commutative algebra, homological algebra, modules and derived categories. Accessed 2026-08-13.

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Quadratic ResiduesGuide · Engineering MathematicsNEXT LESSON →The Law of Quadratic ReciprocityGuide · Engineering MathematicsThe Number Field Sieve and Factoring RecordsGuide · Engineering MathematicsThe Jacobi SymbolGuide · Engineering Mathematics
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