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If p is a prime, the discussion of the congruence x2 ≡ a (p) is fairly easy. It is solvable iff a(p − 1)/2 ≡ (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 x2 ≡ 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.
KeywordsPrime Divisor Arithmetic Progression Quadratic Residue Algebraic Number Field Quadratic Character
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