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

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.

-
This is clearly false, we have $x^{4}+x+1=(x+1) \left(x^3-x^2+x+1\right)$ –  Bombyx mori Dec 25 '12 at 6:53
Your RHS is $x^4+2x+1$. I verified this for $n\leq 100$ in sage. –  JSchlather Dec 25 '12 at 6:54
@user32240 What you have is incorrect. $(-1)^4 + (-1) + 1 = 1$ –  user17762 Dec 25 '12 at 6:55
I see. But I believe now I found one. –  Bombyx mori Dec 25 '12 at 6:56
Sorry ignored the modulo condition given. –  Bombyx mori Dec 25 '12 at 6:58