Find the splitting field of $x^m-1\in\Bbb F_p$.

Question

It is an exercise in Milne's notes. But I don't think I understand the solution...

Here is the solution:

It seems that Milne does not give the justification. So may I please ask for a proof? Or any reference would be helpful. Thanks!

Answer 1

It is a finite field and an extension of $\Bbb F_p$ so it is $\Bbb F_{p^k}$ for some $k$. The $m$-th roots of $1$ are the $m_0$-th roots of $1$ and there are $m_0$ of them. The multiplicative group of $\Bbb F_{p^k}$ contains all $m_0$ of them iff $m_0\mid(p^k-1)$, as the multiplicative group of a finite field is cyclic. So the smallest $k$ that works is the least $k>0$ with $p^k\equiv1\pmod{m_0}$.