If p is a prime, the discussion of the congruence x 2 ≡ a(p) is fairly easy. It is solvable iff a (p-1)/2 ≡ 1 (p). With this fact in hand a complete analysis is a simple matter. However, if the question is turned around, the problem is much more difficult. Suppose that a is an integer. For which primes p is the congruence x 2 ≡ a(p) solvable? The answer is provided by the law of quadratic reciprocity. This law was formulated by Euler and A. M. Legendre but Gauss was the first to provide a complete proof. Gauss was extremely proud of this result. He called it the Theorema Aureum, the golden theorem.
KeywordsArithmetic Progression Quadratic Residue Algebraic Number Field Quadratic Character Legendre Symbol
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