Abstract
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.
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© 1982 Springer Science+Business Media New York
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Ireland, K., Rosen, M. (1982). Quadratic Reciprocity. In: A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics, vol 84. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-1779-2_5
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DOI: https://doi.org/10.1007/978-1-4757-1779-2_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4757-1781-5
Online ISBN: 978-1-4757-1779-2
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