nth root of unity in a cyclic group $\mathbb{Z}_p^*$

Question

Is there a specific set of steps that should be taken in order to the $n$-th root of unity in a cyclic group. To be more specific, I am trying to find the $8$th root of unity for $\mathbb{Z}_{17}^*$. I have already found that $3$ is a generator/primitive root of $\mathbb{Z}_{17}^*$ but I cannot see exactly how this helps find the primitive root.

Answer 1

You are trying to find the elements $x\in\mathbb{Z}_{17}^*$ such that $x^8=1$. In general if you have the equation $x^n=1$ in $\mathbb{Z}_p$ with $p$ prime, there are exactly $d=gcd(n,p−1)$ solutions. One solution is $s=g^{(p-1)/d}$ where $g$ is a primitive element. Then the complete system of solutions is $$\{s^0,s^1,s^2,\cdots,s^{d-1}\}$$ In this case, $p=17$, $d=gcd(8,17-1)=gcd(8,16)=8$ and $g=3$. Then $s=3^2=9$ and the solutions are

• $s^0=1$
• $s^1=9$
• $s^2=9^2=13$
• $s^3=9^3=15$
• $s^4=9^4=16$
• $s^5=9^5=8$
• $s^6=9^6=4$
• $s^7=9^7=2$