Question is to factorise $x^5+x^2+1$ into irreducible polynomials in $\mathbb{Z}$[x].
But I'm fairly sure $x^5+x^2+1$ is already irreducible, but not sure how to prove this as neither Eisenstein's criterion nor showing that it is irreducible mod(p) will work.
The only thing I have so far is to write it as $x^2(x+1)(x^2-x+1)+1$ and show that these are all irreducible, but not sure that that counts.
On
The polynomial $x^5+x^2+1$ is even on the list here of primitive polynomials modulo $2$. This has applications in coding theory. Taking the list of irreducible polynomials of degree $2$ and $3$, one checks that no product of them gives $x^5+x^2+1$.
Show that the polynomial is irreducible over $\mathbb Z_2$. This implies that it is irreducible over $\mathbb Q$