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Proof from first principles of the Ideals generated by $X^{p}-X$ for $p$ a prime

This is exercise $4.2.8$ from Serre’s “A course in Arithmetic” Chapter $2$, if someone could give me an hint.
Let $p$ be a prime. Show that
\langle X^{p}-X \rangle = \{aX\mid a\in\mathbb{Z}\}


The following should give you a hint on how to continue:
If $n$ is a number satisfying $n^{p}=1$, then $n$ must be a power of $2$ (why?). What does this tell you about $X$ and $X^{2}$? (For instance, what do they “look” like?)
If $n$ is a power of $2$, what can you say about $X^{n}$ and $X^{n+1}$? (For instance, what do they “look like”?)
You might also like to try to write out several different cases on paper and see what happens.


I’m afraid it’s not possible to complete the proof by “reduction to cases”.
$X^{p} – X$ is clearly central, so by Lagrange’s theorem the order divides $p$, and thus divides $p-1$ since it’s prime.
Also it is a polynomial of degree $p-1$. It therefore has to split over $F_{p}$ as $X^{p} – X$ has the unique root $1$.
But, as an aside, we can do it by induction. Assume $X^{p} – X$ is the zero polynomial, then $\ldots (X^{p} – X) \ldots = 0$ so $X^{p-1} – 1$ divides $X^{p} – X$. But $X^{p-1} – 1$ is also a polynomial of degree $p-1$, and $X^{p} – X$ must split over $F_{p}$ as $X^{p} – X$ has the unique root $1$.
Note that $X^{p-1} – 1$ is not divisible by $p$.



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