I am reading the group theory text of Eugene Dickson. Theorem 33 shows this polynomial is irreducible
$$ \frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1} \in \mathbb{Z}[x]$$
He shows this polynomial is irreducible in $\mathbb{F}_q[x]$ whenever $p$ is a primitive root mod $q$.
By Dirichlet's theorem there are infinitely many primes $q = a + ke$, so this polynomial is "algebraiclly irreducible", I guess in $\mathbb{Q}[x]$.
Do you really need a strong result such as the infinitude of primes in arithmetic sequences in order to prove this result? Alternative ways of demonstrating this is irreducible for $p$ prime?
COMMENTS Dirichlet's theorem comes straight out of Dickson's book. I am trying to understand why he did it. Perhaps he did not know Eisenstein's criterion. It's always good to have a few proofs on hand.
Another thing is that Eisenstein's criterion is no free lunch since it relays on Gauss lemma and ultimately on extending unique factorization from $\mathbb{Z}$ to $\mathbb{Z}[x]$.