whether $x^4+x^3+x^2+x+1$ is irreducible over $\mathbb Z$ How to check whether $x^4+x^3+x^2+x+1$ is irreducible over $\mathbb Z?$
My guess: If $x^4+x^3+x^2+x+1$ is reducible then $$x^4+x^3+x^2+x+1=f(x)g(x)$$ where $1\le\deg f(x),\deg g(x)\le3$ and $\deg f(x)+\deg g(x)=4$
Checking all the cases for $\deg f(x),\deg g(x)$ is a lengthy exercise. Is there any other way?

If you do not want to use Eisenstein then :
Suppose it is reducible then you have only two choices (Why ??)
$$(x^4+x^3+x^2+x+1)=(x^2+ax+1)(x^2+bx+1)$$
or $$(x^4+x^3+x^2+x+1)=(x^2+ax-1)(x^2+bx-1)$$
This would not take more than two minutes to verify if $a,b$ are in $\mathbb{Z}$ in this case..
