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  1. Quadratic reciprocity - Wikipedia

    • In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: This law, together with its supplements, allows the easy calculation of any Legendre symbol, making it po… See more

    Motivating examples

    Quadratic reciprocity arises from certain subtle factorization patterns involving perfect square numbers. In this section, … See more

    Supplements to Quadratic Reciprocity

    The supplements provide solutions to specific cases of quadratic reciprocity. They are often quoted as partial results, without having to resort to the complete theorem.
    Trivially 1 is a quadratic residue for all primes. Th… See more

    Proof

    Apparently, the shortest known proof yet was published by B. Veklych in the American Mathematical Monthly.
    The value of the Legendre symbol of (used in the proof above) follows directly from Euler's crite… See more

    History and alternative statements

    The theorem was formulated in many ways before its modern form: Euler and Legendre did not have Gauss's congruence notation, nor did Gauss have the Legendre symbol.
    In this article p and q always refer to distinct positive od… See more

    Connection with cyclotomic fields

    The early proofs of quadratic reciprocity are relatively unilluminating. The situation changed when Gauss used Gauss sums to show that quadratic fields are subfields of cyclotomic fields, and implicitly deduced quadratic r… See more

    Other rings

    There are also quadratic reciprocity laws in rings other than the integers.
    In his second monograph on quartic reciprocity Gauss stated quadratic reciprocity for the ring of Gaussian integers, saying that … See more

    Higher powers

    The attempt to generalize quadratic reciprocity for powers higher than the second was one of the main goals that led 19th century mathematicians, including Carl Friedrich Gauss, Peter Gustav Lejeune DirichletSee more

     
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  1. Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x4 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the co...
    en.wikipedia.org/wiki/Quartic_reciprocity
    en.wikipedia.org/wiki/Quartic_reciprocity
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  2. Proofs of quadratic reciprocity - Wikipedia

     
  3. Quartic reciprocity - Wikipedia

  4. Law of Quadratic Reciprocity | Brilliant Math

    In number theory, the law of quadratic reciprocity is a theorem about quadratic residues modulo an odd prime. The law allows us to determine whether congruences of the form x^2 \equiv a x2 ≡ a mod p p have a solution, by …

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  5. Law of Quadratic Reciprocity - ProofWiki

  6. number theory - Uses of quadratic reciprocity theorem

  7. Number Theory - Quadratic Reciprocity - Stanford University

  8. Quadratic reciprocity law - Encyclopedia of Mathematics

  9. Reciprocity law - Wikipedia

  10. elementary number theory - How does law of quadratic reciprocity …

  11. What's the "best" proof of quadratic reciprocity? - MathOverflow

  12. elementary number theory - Euler's Formulation of Quadratic …

  13. Quadratic reciprocity law - Wikipedia

  14. Reciprocity laws - Encyclopedia of Mathematics

  15. Missing link in Wikipedia's proof of quadratic reciprocity law