MHB Find Min Polynomial of $\alpha$ Over $\mathbb{Q} | Solution Included

[kalish1]

I started by setting $\alpha= e^{2\pi i/3} + \sqrt[3]{2}.$ Then I obtained $f(x) = x^9 - 9x^6 - 27x^3 - 27$ has $\alpha$ as a root.

How can I proceed to find the minimal polynomial of $\alpha$ over $\mathbb{Q},$ and identify its other roots?

 

[Opalg]

I started by setting $\alpha= e^{2\pi i/3} + \sqrt[3]{2}.$ Then I obtained $f(x) = x^9 - 9x^6 - 27x^3 - 27$ has $\alpha$ as a root.

How can I proceed to find the minimal polynomial of $\alpha$ over $\mathbb{Q},$ and identify its other roots?

The numbers $e^{2r\pi i/3} + \sqrt[3]{2}e^{2s\pi i/3}$, with $r,s \in\{0,1,2\}$, all satisfy that equation. So that gives you the nine roots of $f(x)$, of which $\alpha$ is one. I'm guessing that $f(x)$ is irreducible over $\mathbb{Q}$ but I don't see how to prove that.