Quadratic reciprocity

For distinct odd primes,

(p / q) (q / p) = (- 1)^{(p - 1) (q - 1) /4}

Quadratic character of minus three

For an odd prime

p \neq = 3

(\frac{- 3}{p}) = (\frac{p}{3}) .

Thus

- 3

p

exactly when

p \equiv 1 (mod 3)

X^{3} + 1 = (X + 1) (X^{2} - X + 1)

shows that

X^{3} = - 1

always has

X = - 1

as a solution. For

p \neq = 3

, it has two further solutions exactly when

- 3

p = 3

, all three factors coalesce at the unique root

- 1

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Quadratic reciprocity is a fundamental theorem in number theory that describes the solvability of quadratic equations modulo prime numbers. It addresses the question of when a quadratic residue exists for two distinct odd prime numbers.