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Math Help - Quadratic Residue proof

  1. #1
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    Quadratic Residue proof

    Show that if a is a quadratic residue (\mod{p}) and ab \equiv 1 (\mod{p}) then b is a quadratic residue (\mod{p}).

    I am not sure where to go with this proof, I started with:
    x^2 \equiv a (\mod{p}) because a is a quadratic residue.
    x^2b \equiv ab (\mod{p})
    x^2b \equiv 1 (\mod{p})

    Stumped right here.

    Then I thought maybe using Legendre's symbols would be better.
    (a/p) = 1 because a is a quadratic residue

    So I have two starting points and no finish line in sight.

    Thanks in advance for any help.

    But not sure where to go to show that (b/p) = 1
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Re: Quadratic Residue proof

    You do know the multiplicative property of the Legendre symbol, right? (That (a/p)(b/p)=(ab/p)).

    Also, (1/p)=1.
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  3. #3
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    Re: Quadratic Residue proof

    Ah I see now. Thanks!

    (a/p) = 1
    (a/p)*(b/p) = 1*(b/p)
    (ab/p) = (b/p)
    (1/p) = (b/p)
    1 = (b/p)
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