$x^6+x^3+1$ is irreducible over $\mathbb{Q}$

Question

I have been trying to prove that $x^6+x^3+1$ is irreducible over $\mathbb{Q}$ (or $\mathbb{Z}$ since by Gauss' Lemma is the same), but I can't. Any idea of how to do so?

Answer 1

HINT: Let $y=x-1$, and apply Eisenstein's criterion for $p=3$.

Answer 2

It's the $9$-th cyclotomic polynomial, and all cyclotomic polynomials are irreducible.

Answer 3

The generalised form of Cohn's criterion also works: all of the coefficients are non-negative and smaller than $2$, and $P(2) = 73$ is prime.