How can I know if the polynomial $x^4 -16x^3 +12x^2 - 3x + 9$ is irreducible over $\mathbb{Z}$?

Question

How can I know if the polynomial $x^4 -16x^3 +12x^2 - 3x + 9$ is irreducible over $\mathbb{Z}$?

I have tried to use Eisenstein's criterion by evaluating on polynomials of the form ax+b but I have not been able to get anywhere. I have also tried to use the Gaussian criterion but I can't come up with anything useful either.

Answer 1

The polynomial is obviously irreducible over $\Bbb F_2$ (or $\Bbb F_5$). By the residue criterion, it is irreducible over $\Bbb Z$.

Edit: by "obviously" I mean, that it has no root in $\Bbb F_2$, and that a comparison of coefficients in $$ x^4-x+1=(x^2+ax+b)(x^2+cx+d) $$ gives equations, which obviously have no solution in $\Bbb Z/2\Bbb Z$.