Show that $x^5-x^2+1$ is irreducible in $\mathbb{Q}[x]$.

Question

Show that $x^5-x^2+1$ is irreducible in $\mathbb{Q}[x]$.

I tried use the Eisenstein Criterion (with a change variable) but I have not succeeded.

Thanks for your help.

Answer 1

$x^5-x^2+1$ is irreducible mod $2$.

This is not too hard to do by hand. Some nice shortcuts are discussed here and here.

Answer 2

$f(1)=1$; $f(-1)=-1$; $f(2)=29$; $f(-3)=-251$; $f(4)=1009$; $f(6)=7741$; $f(10)=99901$.

The above seven examples of $f(x)=$prime or $1$, shows that $f$ is irreducible because if not $f(x)=g(x)h(x)$ and f(x) can not be prime seven times.Do you see why? If not, try it.