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Math Help - the legendre symbol

  1. #1
    Junior Member
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    the legendre symbol

    let p be an odd prime number and let a, b be integers with p does not divide a nor b. Prove that among the congruences x^2= a mod p, x^2= b mod p, and x^2= ab mod p either all three are solvable or exactly one is solvable.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Note  \left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right)\left(\frac{b}{p}\right)

    So now, if  \left(\frac{a}{p}\right) = \left(\frac{b}{p}\right) = 1 , then  \left(\frac{ab}{p}\right) = 1

    if  \left(\frac{a}{p}\right) = \left(\frac{b}{p}\right) = -1 , then  \left(\frac{ab}{p}\right) = 1

    now WLOG assume  \left(\frac{a}{p}\right) = 1 and  \left(\frac{b}{p}\right) = -1 , then  \left(\frac{ab}{p}\right) = -1

    Can you see now?
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