Criteria for Solvability of Certain Congruences

Abstract

Lower case italics will denote rational integers, while lower case Greek letters will denote algebraic integers. The Law of Quadratic Reciprocity can be formulated as follows:If p and a are distinct odd primes, then a is a quadratic residue (mod p) if and only if ( — 1)(p-1)/2p is a quadratic residue (mod q).Let Xp denote the non-principal quadratic character (mod p) and let ζp be a primitive pth. root of unity. Then