Why should a splitting field of a polynomial over $\mathbb{F}_p[X]$ be a cyclotomic extension?

Question

I have just been doing a Galois theory exercise and one part of the exercise requires me to explain why, if $f \in \mathbb{F}_p[X]$ is a polynomial of degree $n$, its splitting field $L$ is a cyclotomic extension of $\mathbb{F}_p$ (i.e. $L/\mathbb{F}_p$ is a cyclotomic extension).

I cannot offer an explanation as to why this is the case. Perhaps it is something to do with there being an element inside the extension with, for example, $x^n = 1$ (thinking of Fermat's little theorem here) but I don't know how to pursue further this line of thinking, or how this is connected to the fact that $L$ is the splitting field of some polynomial.

Answer 1

Since an extension of $\Bbb F_p$ of degree $n$ has order $p^n$ (it's a vector space of dimension $n$ over $\Bbb F_p$), we have that the multiplicative group has order $p^n-1.$

By Lagrange's theorem every element of $\Bbb F_p^*$ is a solution to the equation $$x^{p^n-1}-1=0.$$

Since $L$ is a splitting field, it's the splitting field of $x^{p^n}-x.$

Thus a primitive $p^{n-1}$-th root of unity $\zeta $ will do the trick. That is, $\Bbb F_{p^n}=\Bbb F_p(\zeta) $ is a cyclotomic extension.