A fundamental result in elementary number theory that relates quadratic residuosity between two distinct odd primes p and q via the Legendre symbol: (p/q)(q/p) = (-1)^{((p-1)/2)((q-1)/2)}; it determines whether p is a quadratic residue modulo q in terms of the reverse question up to an explicit sign.

Principle

The organizing rule is a reciprocity symmetry: quadratic character interaction between two odd primes is symmetric apart from a computable sign depending on the congruence classes of the primes modulo 4.

Demonstration

Example: take p=3 and q=11. Compute Legendre symbols (3/11) and (11/3); the law predicts their product equals (-1)^{((3-1)/2)((11-1)/2)} = (-1)^{1*5} = -1, which matches direct computation of quadratic residuosity in each modulus.

Misapplication

Applying the law to include the prime 2 or composite moduli without switching to generalized symbols, or using it without converting to Legendre or Jacobi symbols leads to incorrect residue tests. Treating the law as giving residue values rather than a relation between symbols is also a misuse.

Consequence

Proper use reduces tests of quadratic residuosity between large odd primes to smaller computations and underlies algorithms for primality testing and class field theory; it organizes quadratic residue behavior across primes and enables efficient reciprocity-based evaluations.

Reversal

The reversal is moving to higher power reciprocity laws (cubic, biquadratic, etc.), where symmetry remains but additional auxiliary data and characters are required; those laws are structurally similar but more intricate.

Boundary

Scope: classical statement applies to distinct odd primes and the Legendre symbol; extensions require use of the Jacobi symbol for composites or general reciprocity theorems in number fields. The law does not directly determine residues modulo powers of primes without further work.

Semantic Tension

Tension exists between the Legendre-symbol formulation (primes) and the Jacobi-symbol generalization (composites): a Jacobi symbol value of 1 does not guarantee residuosity, so confusion between the two leads to errors. There is also tension between elementary proofs and deeper field-theoretic interpretations.

Synthesis

The Law of Quadratic Reciprocity is a precise symmetry statement about quadratic characters of odd primes: it expresses how the question 'is p a square mod q?' and its converse are equivalent up to a simple sign determined by residues mod 4, providing a powerful tool for deciding quadratic residuosity and seeding higher reciprocity theories.