Hi everyone, I got stuck in this exercise, where I am to show that the polynomial $\displaystyle g(x)=x^3+x+1$ is irreducible over $\displaystyle \mathbb{Q}[x]$.

In this case you can't use Eisenstein, so what was it that the alternative looked like?

I tried to use polynomial division, assuming it had a root and then tried to reach a contradiction, but that just got me where I started:

$\displaystyle x^3+x+1=(x-a)(x^2+ax+1+a^2)+1+a+a^3$ and then you still had to argue, that $\displaystyle 1+a+a^3$ couldn't be a nonzero, rational number.