Show that $9+9x+3x^3+6x^4+3x^5+x^6$ is irreducible given one of its roots
$\alpha$ is contained in the field $\mathbb{Q}(2^{1/3},\zeta_3)$, where $\zeta_3$ is a primitive cube root of unity. What is the Galois group of that field? How do the Galois automorphisms act on $\alpha$? Hence, how many Galois conjugates (i.e. roots of the same minimal polynomial) does $\alpha$ have?
This uses a little bit of Galois theory. Have you done any?