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In my textbook, it is said that

(Primitive root) Let $p$ be a prime and $n > 1$ be a natural number. The set of all the roots $\alpha$ of the polynomial $x^n - 1 \in Z_p[x]$ forms a cyclic group of order $n$, where all $\alpha$ belongs to the decomposition field of $x^n - 1$ over $Z_p$. If $a$ generates the prementioned group then we call $a$ to be the primitive root or order $n$ over $Z_p$.

I have no idea how to describe the decomposition field of that polynomial over $Z_p$. Please give me an insight to the problem. Thank you

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    $\begingroup$ "decomposition field" aka "splitting field". $\endgroup$ – lhf Nov 11 '18 at 18:55
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    $\begingroup$ As in all finite extensions of $Z_p$, its multiplicative group will be cyclic of order $p^k-1$, where $k$ is the degree of the extension. The decomposition field will be the smallest field for which $n$ divides $p^k-1$. The polynomial is a cyclotomic polynomial over a finite field. See, for example, math.stackexchange.com/questions/305111/…. $\endgroup$ – random Nov 12 '18 at 0:08

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