Why did Gauss prove the same theorem eight times?

Each proof illuminated different structures (Gauss sums, lattice-point counting, cyclotomy), and Gauss sought a proof that would generalize to higher reciprocity laws, which is what later drove the development of algebraic number theory.

What is the difference between the Legendre and Jacobi symbols?

The Legendre symbol is defined for an odd prime modulus and detects quadratic residues exactly; the Jacobi symbol generalizes it to odd composite moduli for computation, but a value of one no longer guarantees the number is a residue.

Quadratic Reciprocity

The law of quadratic reciprocity, which Gauss called the golden theorem, relates whether a prime p is a square modulo q to whether q is a square modulo p, giving a powerful and unexpectedly symmetric criterion for solvability.

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Definition

An integer is a quadratic residue modulo a prime p if it is congruent to a perfect square mod p. Quadratic reciprocity is the theorem relating, for distinct odd primes p and q, the solvability of x squared congruent to q mod p with that of x squared congruent to p mod q.

Scope

This topic covers quadratic residues and nonresidues modulo a prime, Euler's criterion, the Legendre symbol and its multiplicativity, the Jacobi symbol, the two supplementary laws (for minus one and for two), and the main reciprocity law itself, including its role as the first instance of the reciprocity laws of class field theory.

Core questions

Given an odd prime p, which residues are squares, and how does Euler's criterion decide this?
How do the Legendre and Jacobi symbols encode residue information and behave multiplicatively?
What exactly does the reciprocity law assert, and how do the supplements handle minus one and two?
Why is quadratic reciprocity regarded as the prototype of the higher reciprocity laws of class field theory?

Key theories

Euler's criterion and the Legendre symbol: An integer a is a quadratic residue mod an odd prime p exactly when a raised to (p minus one)/2 is congruent to one; the Legendre symbol records this sign and is completely multiplicative in its top argument.
Law of quadratic reciprocity: For distinct odd primes p and q, the product of the two Legendre symbols equals minus one to the power ((p minus one)/2)((q minus one)/2), so reciprocity fails only when both primes are congruent to three mod four.
Supplementary laws and the Jacobi symbol: Separate rules determine when minus one and two are residues, and the Jacobi symbol extends the Legendre symbol to composite moduli, enabling efficient computation without factorization.

Clinical relevance

Reciprocity and the Jacobi symbol give fast algorithms for deciding quadratic residuosity, used in primality tests (Solovay-Strassen), in computing square roots modulo primes, and in cryptographic schemes whose security rests on the quadratic residuosity assumption.

History

Conjectured by Euler and Legendre, the law was first fully proved by Gauss in 1796, who returned to it repeatedly and gave eight different proofs; over two hundred proofs are now known. Its generalization to higher powers motivated Eisenstein, Kummer, and ultimately the reciprocity laws of class field theory.

Key figures

Carl Friedrich Gauss
Adrien-Marie Legendre
Leonhard Euler

Seminal works

irelandRosen1990

Frequently asked questions

Why did Gauss prove the same theorem eight times?: Each proof illuminated different structures (Gauss sums, lattice-point counting, cyclotomy), and Gauss sought a proof that would generalize to higher reciprocity laws, which is what later drove the development of algebraic number theory.
What is the difference between the Legendre and Jacobi symbols?: The Legendre symbol is defined for an odd prime modulus and detects quadratic residues exactly; the Jacobi symbol generalizes it to odd composite moduli for computation, but a value of one no longer guarantees the number is a residue.