this question is from the book Graduate Algebra written by Lang.
It asks me to find the splitting field of $f(x)=x^{p^{8}} -1$ over the field $Z_{p}$.
I got a proof from my friend but I don't really understand it.
So the proof is:
Notice that $x^{p^{8}} -1 = (x-1)^{p^{8}}$ in $Z_{p}$, so we can linearly factor f(x) into $(x-1)^{p^{8}}$ and all the $1$ in the linear factors are in $Z_{p}$, so $Z_{p}$ is the splitting field.
I understand the second half part of the proof, since it is the definition of the splitting field, but why could I get $x^{p^{8}} -1 = (x-1)^{p^{8}}$ in $Z_{p}$.
If the proof is not correct, could you give me some hints or detailed proofs?
Thank you!
It was incorrect...but close:
$$x^{p^8}-1=\left(x^p-1\right)^{p^7}\;\ldots$$