List of Quadratic Reciprocity Proofs

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Many proofs of quadratic reciprocity are presented here.

Theorema Fundamentale in Doctrina de Residuis Quadraticis.

\displaystyle \left(\frac{-1}{q}\right)=(-1)^{\frac{q-1}{2}}

\displaystyle \left(\frac{2}{q}\right)=(-1)^{\frac{q^{2}-1}{8}}

\displaystyle \left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}.\frac{ q-1}{2}}

Gauss has proof by induction by reducing the problem of computing to Legendre symbol for a particular pair of primes to smaller numbers. Being a residue is captured by an equation, but what about non-residue? Gauss ingenious idea is to use auxiliary primes, to convert non-residue property into residue property of a product of primes.

Proofs using Gauss Lemma: Study how multiplication with one prime acts on a segment modulo other prime( half the classes or even classes etc.) This give a formula for the residue symbol.

Zolotarev’s lemma: Also studies multiplication by one prime and the permutation structure of this map.

Eisenstein’s Lattice point proof. Also studies multiplication by primes on the even class modulo other prime. Once we get a formula for the residue symbol, the relation amounts to counting lattice points.

Analytic proofs with sine function: The sine function captures the effect of Gauss’s lemma about number of elements falling outside the segment.

Proofs by Gauss sums: Roughly this allows us to get an expression for square root of a prime p in terms of roots of unity. Then deciding if that prime p \mod q is square modulo other prime q is done by Euler’s criterion – hence we need to rise the Gauss sum to the power q – but then we see that q \mod p comes into the picture.

Algebraic Number Theory: Proof by Gauss sums can be seen as coming from more general ideas. Raising to power q is the Frobenius element and what we have a description of Frobenius elements. Galois groups, cyclotomic fields, quadratic fields.

Proofs by Poisson Summation, Theta functions: This directly allow us to relate expression with p/q in terms of expressions with q/p

Proofs by Quadratic Forms: 1. Use reduction, composition, genus theory of quadratic forms.
2. Use existence of solutions to Pell’s equation.
3. Using Hilbert symbols

Using Jacobi sums and counting solutions to equations:

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