The Legendre symbol, including its zero value

Depends on

• Quadratic residues and nonresidues modulo an integer

Used by

• x²≡ a (mod p) has exactly 1+(a/p) solution classes Corollary
• A soluble square congruence need not define a quadratic residue Counterexample
• The Jacobi symbol, with its zero value and empty-product convention Definition
• A complete reciprocity table for 3,5,7,11 Example
• A nonmonic quadratic congruence solved through its discriminant Example
• Odd primes congruent to 1 or 3 modulo 8 are represented by x²+2y² Example
• Odd primes represented by a divisor of x²+3 Example
• Quadratic residues, roots, and the Legendre table modulo 11 Example
• A prime congruent to 3 modulo 4 divides both coordinates of a divisible two-square sum Lemma
• On the units, the Legendre symbol is the unique nontrivial homomorphism to {±1} Proposition
• The Legendre symbol is well defined on residue classes Proposition
• Euler's criterion: (a/p)≡ a^(p-1)/2 (mod p) Theorem
• Fermat's two-square theorem for primes Theorem
• First supplement: (-1/p)=(-1)^(p-1)/2 Theorem
• Gauss's quadratic-residue lemma Theorem
• The Legendre symbol is multiplicative for all integer numerators Theorem
• The odd-prime Hilbert symbol formula Theorem
• Unit square criterion and root count modulo odd prime powers Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• H. Hackman, Elementary Number Theory, Chapter D, Section D.I (standard reference, not scraped)
• W. Stein, Elementary Number Theory, Section 4.1 (standard reference, not scraped)
• A. Gorodnik, Number Theory, Lecture 9, Section 1 (standard reference, not scraped)