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Math Help - primitive roots

  1. #1
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    primitive roots

    Let g be a primitive root of an odd prime p. Show that -g is also a primitive root of p if and only if p is congruent to 1 mod 4.
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    Quote Originally Posted by NikoBellic View Post
    Let g be a primitive root of an odd prime p. Show that -g is also a primitive root of p if and only if p is congruent to 1 mod 4.

    -g is a prim. root of p iff -1=(-g)^\frac{p-1}{2}=(-1)^\frac{p-1}{2}g^\frac{p-1}{2}=(-1)^\frac{p-1}{2}(-1)\Longleftrightarrow (-1)^\frac{p-1}{2}=1 \Longleftrightarrow \frac{p-1}{2}=2k\,,\,\,k\in\mathbb{Z} \Longleftrightarrow ...

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