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Primitive root

  1. Nov 19, 2013 #1
    1. The problem statement, all variables and given/known data
    p prime, If p=1 ( mod 3) then Zp contains primitive cube roots of unity. Now I am considering which p does Zp contains primitive fourth roots of unity.

    opposite way? I mean if p=1(mod4) then Zp contains primitive fourth roots of unity??

    2. The attempt at a solution
    I can prove that if Zp contains primitive fourth roots of unity, then 4|(p-1) . but how about the opposite way? I mean if p=1(mod4) then Zp contains primitive fourth roots of unity?? I know this statements true if q prime instead of 4. And what values of p does Zp contains primitive fourth roots of unity???
     
  2. jcsd
  3. Nov 19, 2013 #2

    I like Serena

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    Hi Funky1981! :smile:

    The expression ##p \equiv 1 \pmod 3## means that there is a k such that ##p=3k+1##.

    Now suppose g is a primitive root mod p.
    Then ##g^{\phi(p)} \equiv g^{3k} \equiv 1 \pmod p##.
    Therefore ##g^k## is a cube root of 1 in ##\mathbb Z_p##.


    Same argument holds for ##p \equiv 1 \pmod 4##...
     
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