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Math Help - primitive root of a prime

  1. #1
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    primitive root of a prime

    Hi,

    let p be a prime congruent to 1 mod 3
    let a be a prim root of p
    suppose a is not congruent to 0 mod p

    show that there is no solution x for the congruence x^2 = a mod p


    i know by some theorem about primitive roots of primes that x^2 - a = 0 mod p has at most 2 solutions

    i also know that the order of a modulo p is phi(p) = p -1, and so a^(p - 1) = 1 mod p, and p - 1 = 3k for some k, since p = 1 mod 3.

    anyone have any hints? i started also by noting that (x^2)^(p - 1) = a^(p -1) mod p, and tried factoring some stuff with that, that is after assuming for a contradiction that in fact there is a solution x
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    x^{p-1}\equiv 1 \mod p holds for all x relatively prime to p, not just for primitive roots. So if x^2 \equiv a \mod p, what do you get if you raise both side to the power (p-1)/2?
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