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Math Help - Primitive Roots

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    Primitive Roots

    Show that if g and h are primitive roots of an odd prime p, then their product gh is not a primitive root of p.
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    Quote Originally Posted by tarheelborn View Post
    Show that if g and h are primitive roots of an odd prime p, then their product gh is not a primitive root of p.

    Hint: x is a primitive root of p iff x^{\frac{p-1}{2}}=-1

    Tonio
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by tonio View Post
    Hint: x is a primitive root of p iff x^{\frac{p-1}{2}}=-1

    Tonio
    That is not true. For instance -1 is certainly not a primitive root for any prime p>3, but (-1)^{(p-1)/2}=-1 for primes of the form p=4n+3...

    What you mean is that if x is a primtive root then x^{(p-1)/2}=-1, i.e. if x is a primitive root then it's not a quadratic residue. Since the product of two nonresidues is a residue...
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    I am sorry, but I am not following you. Can you please elaborate?
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    MHF Contributor Bruno J.'s Avatar
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    If x,y are primitive roots, then x^\frac{p-1}{2}=y^\frac{p-1}{2}=-1, and (xy)^\frac{p-1}{2}=(-1)^2=1, therefore xy is not a primitive root.
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    Quote Originally Posted by Bruno J. View Post
    That is not true. For instance -1 is certainly not a primitive root for any prime p>3, but (-1)^{(p-1)/2}=-1 for primes of the form p=4n+3...

    What you mean is that if x is a primtive root then x^{(p-1)/2}=-1, i.e. if x is a primitive root then it's not a quadratic residue. Since the product of two nonresidues is a residue...

    Of course. Gross typo there: if w is a primitive root then w^{\frac{p-1}{2}}=-1. Thanx

    Tonio
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