Quadratic, Cubic, Biquadratic, and Quintic Reciprocity
Darrell Cox, Sourangshu Ghosh, Eldar Sultanow · International Journal of Pure and Applied Mathematics Research · 2022
IntroductionLet p be an odd prime.If the congruence 2x n (mod p) has a solution, we say that n is a quadratic residue mod p and write nRp.If the congruence has no solution we say that n is a quadratic nonresidue mod p and write nRp.If 0 n (mod p) we define Legendre's symbol (n|p) as follows: (n|p) = +1 if nRp or -1 if nRp.If n = 0(mod p) we define (n|p) = 0.The quadratic reciprocity law (first proved by Gauss) states that if p and q are distinct odd primes, then (p|q) = (q|p) unless p q 3(mod 4), in which case (p|q) = -(q|p).