# Decomposition field of $x^n - 1$ over $Z_p$

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

• "decomposition field" aka "splitting field". – lhf Nov 11 '18 at 18:55
• 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/…. – random Nov 12 '18 at 0:08