The Jacobi Symbol and a Method of Eisenstein for Calculating It
نویسنده
چکیده
We present an exposition of the basic properties of the Jacobi symbol, with a method of calculating it due to Eisenstein. Fix a prime p. For an integer a relatively prime to p the Legendre symbol is defined by (a/p) = 1 if a is a quadratic residue (mod p) and (a/p) =−1 if a is a quadratic nonresidue (mod p). We recall Euler’s theorem that (a/p) ≡ a(p−1)/2 (mod p). We have the famous Law of Quadratic Reciprocity: Theorem 1. (The Law of Quadratic Reciprocity) Let p and q be distinct odd primes. Then
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