<s>[INST] 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?
 [/INST] Substitute $\;x\to x+1\;$ , which in fact is an automorphism of the ring of pol's:
$$x^4+x^3+x^2+x+1\to x^4+5x^3+10x^2+10x+5$$
Now just apply Eisenstein with $\;p=5\;$ to the above right polynomial...
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