= Solution
{c}
Reduce the monic integer <polynomial> modulo $p$. Every $u\in\mathbb F_p$ satisfies $u^p=u$, so $u^p-u-1=-1\ne0$. Thus the reduction has no <root> in $\mathbb F_p$. If it had a proper factor, part 2 would force such a <root>, a contradiction. Hence it is <irreducible> over $\mathbb F_p$.
The criterion of <irreducibility by degree-preserving reduction modulo a prime>, which follows from <Gauss's lemma>, now applies because the <polynomial> is monic and its <degree> is preserved on reduction. Consequently \b[$X^p-X-1$ is <irreducible> in $\mathbb Q[X]$].
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