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Given a polynomial $f(x)=9+9x+3x^3+6x^4+3x^5+x^6$ and one of its roots $\alpha=2^{1/3}+e^{2\pi i/3}$. Show that $f(x)$ is irreducible in $\mathbb Q[x]$.

Eisenstein's criterion fails, it also didn't use the fact that $\alpha$ is a root of $f(x)$. How should I approach? Thanks

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$\alpha$ 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?

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