For any prime $p$ and any non-zero $a \in \mathbb{F}_p$, prove that $f = x^p-x+a$ is irreducible and seperable.
Now, I believe you can show irreducibility of $f$ this way:
$f \in \mathbb{F}_p[x]$ is irreducible
$$\iff$$
$\operatorname{Gal}(\mathbb{F}_p)$ acts transitively on the roots of $f$.
Specifically, by looking at the frobenius endomorphism $x \mapsto x^p$ in $$\operatorname{Gal}(\mathbb{F}_p(r)/\mathbb{F}_p)$$
and since $\mathbb{F}_p(r)$ I believe is finite, we have that the frobenius endomorphism is surjective, hence an isomorphism, hence in $$\operatorname{Gal}(\mathbb{F}_p(r)/\mathbb{F})$$ and one finds that it act´s transitively on the roots, which in thise case are $r,r+1,\ldots,r+(p-1)$ for some root $r$ of $f$. Since it means that $f$ is irreducible in $\mathbb{F}_p(r)[x]$ it must be irreducible in $\mathbb{F}_p[x]$ (if I am not mistaken).
Now, I believe there should be a way of showing that $p$ is irreducible without relying on galois.
In particular, you should be able to look at derivatives of $p(x)$. But I am finding it very hard.
Any hints/or solutions would be welcome.
To be clear, I believe I can show seperable, so this question is specifically asking about the irreducibility.
