If $a=\sqrt[3]{2+\sqrt{6}}$ then $a^3 - 2 = \sqrt{6}$ so $a^6 - 2a^3 - 2 = 0$. Thus, $a$ is root of polynomial $x^6 - 2x^3 - 2$. This polynomial is irreducible so $\mathbb{Q}(a)$ are all elements of form $a_0 + a_1 a+ ... + a_5a^5$ for $a_i \in \mathbb{Q}$.