Irreductibility of a polynom

Question

I have to show that $p(x)=x^5+5x+11$ is irreductible on $\mathbb{Z}[x]$ both using and not using Eisenstein. I have been trying to translate $p(x)$ or searching a field $\mathbb{Z}p$ but can't reach anywhere even with Eisenstein

Answer 1

You can use Eisensteins criterion on $p(x-1)$. Without using Eisenstein, you could note that $$p(x)\equiv x^5+2x+2\pmod{3},$$ which is irreducible as it has no roots in $\Bbb{F}_3$, and the three irreducible quadratics $x^2+x+2$, $x^2+2x+2$ and $x^2+1$ do not divide it.