I need to find the splitting field of a polynomial $ x^m-1 \in\mathbb{F}_p[x] $.
I know that if $ \gcd(m,p)=1 $ then the splitting field is $\mathbb{F}_p(z)$ where $ z $ is primitive root of unity of order $ m $. My question is what to do if $ \gcd(m,p) \ne 1 $. How can I find the splitting field then?
Thanks for the helpers.
The general case reduces to the one where $m$ and $p$ are coprime by noting that if they are not, then $m$ is actually a multiple of $p$, and we use the identity $x^{np} - 1 = (x^n - 1)^p$ which holds since the characteristic is $p$.
By writing $m = np^k$ for $n < p$ and iterating the above, we get that $x^m - 1$ is just a power of $x^n - 1$, so these have the same splitting field.