Minimal polynomial of $\sqrt{3+\sqrt[3]{3}}$

Question

Given a real number $\alpha:=\sqrt{3+\sqrt[3]{3}}$, I've got to determine the minimal polynomial of $\alpha$ over the rational numbers $\mathbb{Q}$. I know that squaring results in: $\alpha^2=3+\sqrt[3]{3}$. Now I don't know how to go on because of $\sqrt[3]{3}$, that is obviously not included in $\mathbb{Q}$. Is there any general strategy to find a minimal polynomial?

Answer 1

You just continue with the algebraic manipulation to find $$(x^2 - 3)^3 - 3 = x^6 - 9 x^4 + 27 x^2 - 30$$

and this is the minimal polynomial as it is irreducible by Eisenstein with $p=3$: $p | 9,27,30$ and $p^3 \not | 30$.