Law of Quadratic Reciprocity

 

Law of Quadratic Reciprocity

The Law of Quadratic Reciprocity is a fundamental result in number theory that relates the solvability of certain quadratic equations to the primes that divide their coefficients. The statement of the law is as follows:

Let p and q be distinct odd prime numbers. Then:

If p ≡ 1 (mod 4) or q ≡ 1 (mod 4), then the equation x^2 ≡ q (mod p) has a solution if and only if the equation x^2 ≡ p (mod q) has a solution.

If p ≡ q ≡ 3 (mod 4), then the equation x^2 ≡ p (mod q) has a solution if and only if q divides (p-1)/2.

The Law of Quadratic Reciprocity was first conjectured by the mathematician Euler and later proved by Gauss using the theory of quadratic residues. In this proof, we will use a different approach that is based on the properties of the Legendre symbol.

Let (a/p) denote the Legendre symbol, which is defined as follows:

(a/p) = 0 if a ≡ 0 (mod p)

(a/p) = 1 if the equation x^2 ≡ a (mod p) has a solution

(a/p) = -1 if the equation x^2 ≡ a (mod p) has no solution

Using the properties of the Legendre symbol, we can prove the Law of Quadratic Reciprocity as follows:

If p ≡ 1 (mod 4) or q ≡ 1 (mod 4), then (q/p) = (p/q).

Proof: Without loss of generality, assume that p ≡ 1 (mod 4). Then we have:

(q/p) = (-1)^((p-1)/2 * (q-1)/2) * (p/q)

= (p/q)

since (-1)^((p-1)/2 * (q-1)/2) = 1. Thus, (q/p) = (p/q).

If p ≡ q ≡ 3 (mod 4), then (q/p) = -(p/q).

Proof: Without loss of generality, assume that p ≡ 3 (mod 4) and q ≡ 3 (mod 4). Then we have:

(q/p) = (-1)^((p-1)/2 * (q-1)/2) * (p/q)

= -(p/q)

since (-1)^((p-1)/2 * (q-1)/2) = -1. Thus, (q/p) = -(p/q).

Using these two properties of the Legendre symbol, we can prove the Law of Quadratic Reciprocity as follows:

If p ≡ 1 (mod 4) or q ≡ 1 (mod 4), then x^2 ≡ q (mod p) has a solution if and only if x^2 ≡ p (mod q) has a solution.

Proof: By the first property of the Legendre symbol, we have (q/p) = (p/q). Thus, we have:

x^2 ≡ q (mod p) has a solution

if and only if (q/p) = 1

if and only if (p/q) = 1

if and only if x^2 ≡ p (mod q) has a solution

since (a/p) = 1 if and only if the equation x^2 ≡ a (mod p) has a solution.

If p ≡ q ≡ 3 (mod 4), then x^2 ≡ p (mod q) has a solution if and only if q divides (p-1)/2.

Proof: By the second property of the Legendre symbol, we have (q/p) = -(p/q). Thus, we have:

x^2 ≡ p (mod q) has a solution

if and only if (q/p) = 1

if and only if (p/q) = -1

if and only if q divides (p-1)/2

since (a/p) = -1 if and only if the equation x^2 ≡ a (mod p) has no solution.

Therefore, we have proved the Law of Quadratic Reciprocity using the properties of the Legendre symbol. This proof is elegant and straightforward, and it has the advantage of avoiding the more advanced concepts from algebraic number theory that are used in other proofs of this theorem.