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Math Help - Primitive Roots of Unity and GCD

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    Primitive Roots of Unity and GCD

    We say a complex number z is a primitive nth root of unity if  z^n=1 but  z^m\neq1 for 0<m<n. Show that  \omega^k is a primitive nth root of unity if and only if gcd(k,n)=1.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by meggnog View Post
    We say a complex number z is a primitive nth root of unity if  z^n=1 but  z^m\neq1 for 0<m<n. Show that  \omega^k is a primitive nth root of unity if and only if gcd(k,n)=1.
     (\Longrightarrow) Assume  g=(a,b)>1 \implies \frac ng < n and suppose  k=gd .  \left(\omega^k\right)^{n/g} = \omega^{dgn/g} = \left(\omega^n\right)^d = 1 . Thus  \omega^k is not primitive.

     (\Longleftarrow) If  (k,n)=1 and  i<n , then  ki=dn+r where  0\leq r<n . Thus  \omega^{ki} = \omega^{dn}\omega^r = \omega^r\neq1 . So we see  \omega^k is primitive.
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