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.

Hint: every finite field $F$ has a cyclic multiplicative group $F^\times$. Does that give an idea for a polynomial that every element of $F$ must be a root of? — Greg Martin, May 04 '24 at 20:32
This previous Question Finite extensions of finite fields always generated by cyclotomic type polynomials might help you tie out your intuition. — hardmath, May 04 '24 at 20:37
@GregMartin That in, for example, $\mathbb{F}p$ every element is a $p$-th root of 1? So essentially is it true in more generally that $\mathbb{F}{p^n}/\mathbb{F}_p$ is a cyclotomic extension (by what you have said) and this implies the result in the exercise I was doing? — Featherball, May 04 '24 at 21:06
See this Question (among others) for the proposition that the multiplicative group of a finite field is cyclic. — hardmath, May 04 '24 at 21:11
Basically this is because every non-zero element of every algebraic extension of $\Bbb{F}_p$ is a root of unity of some order. — Jyrki Lahtonen, May 04 '24 at 21:14
@JyrkiLahtonen Plus the fact that there is a primitive element. — , May 05 '24 at 01:59

suckling pig · Answer 1 · 2024-05-05T02:08:11.330

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.

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

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