What is the smallest Finite Field in which the following polynomial is factorizable to irreducible factors?

Question

What is the smallest Finite Field in which the following polynomial is factorizable to irreducible factors?

$$(x^2+x+1)(x^5+x^4+1)(x^7+x^6+x^3+1) $$

Answer 1

There is no finite field in which all given three polynomials are irreducible. The last polynomial, i.e., $x^7+x^6+x^3+1$ has always $-1$ as a root, hence is reducible over all finite fields.
On the other hand, $\mathbb{F}_3$ is the smallest finite field, in which all three polynomials are reducible: $x^2+x+1=(x+2)^2$, $x^5+x^4+1=(x^3 + 2x + 1)(x + 2)^2$, and $x^7+x^6+x^3+1=(x^6 + x^2 + 2x + 1)(x + 1)$.