Irreducibility of $x^{n}+x+1$

Question

Motivated by this problem, and KCd's comment on my answer, I am left with the following question:

Question: Suppose that $n\not \equiv 2\pmod{3}$. Is $$x^n+x+1$$ irreducible over $\mathbb{Q}$?

I am not sure how to solve this, any thoughts are appreciated.

Answer 1

Yes, it is true. See the second claim of Theorem 1 on page 289.