Irreducibility of $~\frac{x^{6k+2}-x+1}{x^2-x+1}~$ over $\mathbb Q[x]$

polynomials irreducible-polynomials factoring cyclotomic-polynomials

The Artin—Schreier polynomial $~x^n-x+1~$ is always irreducible over $\mathbb Q[x]$, unless $n=6k+2$, in which case it seems to have only two factors, one of which is always $x^2-x+1$. The irreducibility of its other 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$\big)$. Thank you.


Solution 1:

Yes, see Theorem 3 in this paper by Ljunggren.

Related

Elementary proof that monotone functions are differentiable somewhere
Parts with the maximum number of each integer
Can algebraic numbers be compared using only rational arithmetic?
Computing ${\int\limits_{0}^{\chi}\int\limits_{0}^{\chi}\int\limits_{0}^{2\pi}\sqrt{1- \cdots} \, d\phi \, d\theta_1 \, d\theta_2}$?
Learning Math Efficienctly and Succeeding in Grad School
Can you define arc length using a piece of string?
Does this sequence always terminate or enter a cycle?
Why does this pattern of "nasty" integrals stop?
Smooth functions for which $f(x)$ is rational if and only if $x$ is rational
Geometric intuition for the Weingarten map
What is the probability that a random walk on $\mathbb{Z}^2$ will hit $(1,0)$ before $(2,0)$?
Does Birkhoff - von Neumann imply any of the fundamental theorems in combinatorics?