I want to determine the minimal polynomial of $a=\sqrt{2+\sqrt[3]{2}}$ over $\mathbb{Q}$.
I tried:
$a=\sqrt{2+\sqrt[3]{2}} \Rightarrow a^2=2+\sqrt[3]{2} \Rightarrow (a^2-2)^3=2 \Rightarrow a^6-6a^4+12a^2-10=0$
So $a$ is a root of the polynomial $f(a)=a^6-6a^4+12a^2-10$.
With Eisenstein's criterion and $p=2$, $f(a)$ is irreducible in $\mathbb{Z}[x]$ and also in $\mathbb{Q}[x]$.
$\Rightarrow f(a)$ is the minimal polynomial of $a$ over $\mathbb{Q}$.
Is this way right?