TO check for irreducibility of given polynomial

Question

$x^6+x^3+1$ is irreducible over $\mathbb{Q}[x]$.i know it cannot have any linear factor in $\mathbb{Q}[x]$ since 1 and -1 doesnot satisfied it.but how do i rule out other possibilities of degree 2,4 and of degree 3,3???

Answer 1

At your place I would be way more direct and set $t=x-1$ and apply Eisenstein's criterion (to this new polynomial in the formal variable $t$) with $p=3$. This rules out all factor possibilities. ;-)

But if you want a more elementary way, I would suggest remarking that your polynomial $P(x)$ is but $Q(x^3)$ with $Q(x)=x^2+x+1$ which has roots $j$ and $j^2$, and I would examinate cubic roots of $j$ and $j^2$.