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Math Help - Primitive root

  1. #1
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    Primitive root

    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.

    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???
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  2. #2
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    Re: Primitive root

    Hi,
    By a primitive 4th root of unity, I assume you mean a y such that y4 is 1 mod p, but no smaller power is 1. Now if you know about cyclic groups, your problem is easy. However, even without this here's a proof.

    There is a primitive 4th root of unity mod p if and only if 4 divides p-1.
    Proof of only if:
    Let x be a primitive root mod p; i.e. every non-zero element of Zp is a power of x. (You need to know such exist.) So suppose y=xm is a primitive 4th root. I need the following fact: the order of xm mod p is (p-1)/gcd(m,p-1). (Order is the least positive power k with (xm)k = 1 mod p; this is a standard result from cyclic groups. But it's not hard to prove separately.) So p-1=4gcd(m,p-1) or 4 divides p-1. QED.
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