On quadratic residues

Abstract

is identical with Jacobi's symbol (m/n) and obtain a proof of the quadratic reciprocity theorem. If m and n have a common factor g, then A= O, since the numerator of A has the factor Sqm _ S-qm = o, where q = n / g. 2. We shall show that, when m and n are relatively prime, A is independent of the particular primitive root s employed. First, let m be positive and odd, and let r be a primitive mth root of unity. Since m is odd, the mth roots of unity are 1 n r2 r4, . . ., r2(-l) . Hence