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What is quadratic reciprocity

  1. Jul 23, 2014 #1
    Definition/Summary

    A number n is a quadratic residue mod m if there exists some number a which, squared mod m, gives n.

    Equations

    Definition of the Legendre symbol, for any number a and for any odd prime p:
    [tex]\left(\frac ap\right)=\begin{cases}
    0&p|a\\
    1&\exists n:n^2\equiv a\pmod p\\
    -1&\nexists n:n^2\equiv a\pmod p
    \end{cases}[/tex]

    The Legendre symbol is multiplicative:
    [tex]\left(\frac{ab}{p}\right)=\left(\frac ap\right)\left(\frac bp\right)[/tex]

    The Law of Quadratic Reciprocity, for any odd primes p and q:
    [tex]\left(\frac qp\right)=(-1)^{(p-1)(q-1)/4}\left(\frac pq\right)[/tex]

    Extended explanation

    For example, 0 1 4 5 6 and 9 are quadratic residues mod 10 because the squares of "ordinary" numbers (which are "base 10") can end in 0 1 4 5 6 or 9.

    2 3 7 and 8 are not quadratic residues mod 10.

    The law of Quadratic Reciprocity, of course, does not apply mod 10, because 10 is not a prime.

    A generalisation of the Legendre symbol for odd non-primes p is the Jacobi symbol.

    There is also a Hilbert symbol.

    There are extensions of the law of Quadratic Reciprocity for non-prime p and q.

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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