Quadratic Reciprocity

  • Kenneth Ireland
  • Michael Rosen
Part of the Graduate Texts in Mathematics book series (GTM, volume 84)


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.


Prime Divisor Arithmetic Progression Quadratic Residue Algebraic Number Field Quadratic Character 
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Copyright information

© Springer Science+Business Media New York 1982

Authors and Affiliations

  • Kenneth Ireland
    • 1
  • Michael Rosen
    • 2
  1. 1.Department of MathematicsUniversity of New BrunswickFrederictonCanada
  2. 2.Department of MathematicsBrown UniversityProvidenceUSA

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