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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
$$
\langle X^{p}-X \rangle = \{aX\mid a\in\mathbb{Z}\}
$$

A:

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.

A:

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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