Irreducibilidad de$~\frac{x^{6k+2}-x+1}{x^2-x+1}~$ sobre$\mathbb Q[x]$

Question

El polinomio Artin-Schreier $~x^n-x+1~$ es siempre irreducible sobre$\mathbb Q[x]$, a menos que$n=6k+2$, en cuyo caso parece tener solo dos factores, uno de los cuales es siempre$x^2-x+1$. La irreductibilidad de su otro factor, $$\dfrac{x^{6k+2}-x+1}{x^2-x+1}$$ holds for all k lesser than $790$, at the very least. My question would be whether it holds $~ \ forall ~ k \ in \ mathbb N$.
$\ big ($I have no formal training in abstract algebra, other than knowing the high-school definitions
of groups and rings: that's it. I mention this in case you are probably wondering by now about
the near-lack of any meaningful ideas, on my side, about how to even approach this problem.
I realize that I am in over my head, but the question is so interesting, that I simply could not
resist the temptation, and just had to ask it. Hope you will not hold it against me$\ big)$ . Gracias.

Answer 1

Sí, consulte el Teorema 3 en este artículo de Ljunggren .