PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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The Legendre symbol is well defined on residue classes

Statement

For every odd prime $p$ , the Legendre symbol belongs to ${- 1, 0, 1}$ , depends only on the numerator modulo $p$ , and satisfies

$(\frac{a}{p}) = 0 ⟺ p ∣ a .$

Facts & Assumptions

Given: An odd prime $p$ and integers $a, b$ with $a \equiv b (m o d p)$ .

[L1]

The Legendre symbol is $0$ on a numerator divisible by $p$ , $1$ on a quadratic residue modulo $p$ , and $- 1$ on a quadratic nonresidue (The Legendre symbol, including its zero value).

[L2]

The congruence $a \equiv b (m o d p)$ means that $p$ divides $a - b$ (Congruence modulo an integer: $a \equiv b (m o d n)$ when $n ∣ (a - b)$ , including the moduli $0$ and $1$ ).

[L3]

Quadratic residuosity of a unit integer depends only on its residue class (Quadratic residuosity is representative-independent and the residues are the image of squaring).

Proof

technique · direct

1.1

By [L2], $p ∣ (a - b)$ , so $p ∣ a$ exactly when $p ∣ b$ . Thus congruent numerators enter the zero branch of [L1] simultaneously.

L1L2given

2.1

If $p ∤ a$ , then also $p ∤ b$ by step 1.1, and [L3] says that $a$ and $b$ are simultaneously quadratic residues or simultaneously nonresidues. Hence [L1] assigns them the same sign.

L1L3step 1.1

3.1

The three disjoint branches in [L1] give only the values $- 1, 0, 1$ ; step 1.1 proves that divisibility gives value zero, and the two unit branches give nonzero values. Therefore the symbol is representative-independent and is zero exactly when $p$ divides its numerator.

L1step 1.1step 2.1∎

Depends on

Used by

On the units, the Legendre symbol is the unique nontrivial homomorphism to {±1} Proposition

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 30 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

H. Hackman, Elementary Number Theory, Chapter D, Section D.I (standard reference, not scraped)
A. Gorodnik, Number Theory, Lecture 9, Section 1 (standard reference, not scraped)