We know from Moore's theorem and the construction of finite fields that $\mathbb{F}_{p^n}$ is the splitting field of $X^{p^n}-X$ over $\mathbb{F}_p$. I was wondering what the $X^{p^n}-1$ splitting field would be and I imagine this is related to cyclotomic extensions, but we haven't studied those yet in my fields and Galois theory course.
I thought about looking at $X^{p^n}-X$ as a factor of a polynomial of the form $X^{p^m}-X$, but I wanted to know the canonical way to proceed.
Theorem (Kummer) Let $p$ be a prime, and $m,n$ be positive integers. The highest power of $p$ that divides $\binom{n+m}{n}$ is the number of carries when adding $n$ and $m$ in base $p$.
Corollary. Let $p$ be a prime, $n\geq 1$, and let $k$ be an integer, $1\lt k\lt p^n$. Then $p\mid \binom{p^n}{k}$.
Proof. There is at least one carry when adding $p^n-k$ and $k$ in base $p$, since both $p^n-k$ and $k$ have fewer than $k+1$ digits in base $p$, but their sum has $k+1$ digits. $\Box$
Corollary. If $F$ has characteristic $p$, and $a,b\in F$, then $(a+b)^{p^n} = a^{p^n}+b^{p^n}$.
Corollary. In $\mathbb{F}_{p^k}[x]$, for all $r\geq 1$ we have $x^{p^r}-1 = (x-1)^{p^r}$. In particular, $x^{p^r}-1$ splits over any field of characteristic $p$.
So the splitting field over $\mathbb{F}_p$ is $\mathbb{F}_p$ itself.
