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Math Help - Amount of quadratic residues and nonresidues modulo p

  1. #1
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    Amount of quadratic residues and nonresidues modulo p

    I cannot find a proof/ reasoning behind the following statement:

    If p is an odd prime then among the integers in [1,p-1] there are exactly \frac{p-1}{2} quadratic residues and exactly \frac{p-1}{2} nonresidues.

    My text says the reasoning behind this follows from the following fact:

    Let g be a primitive root modulo p and assume \gcd(a,p)=1. Let r be any integer such that g^r\equiv{a}(mod\,p). Then r is even if and only if a is a quadratic residue modulo p.

    But I don't really see how that follows from this. Any help would be appreciated.

    Thanks.
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Re: Amount of quadratic residues and nonresidues modulo p

    There is a proof on this page: Quadratic Residues
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