Splitting Field of $x^6-3x^4+3x^2-3$ \ Normal Extension \ Subfields

Question

i'm trying to solve the problem above. Let's name L the splitting field. Using $x^2=t$, i have found 6 roots $$\alpha=\sqrt{1+\sqrt[3]{2}} \\ \beta=\sqrt{1+\varepsilon\sqrt[3]{2}} \\ \gamma=\sqrt{1+\varepsilon^2\sqrt[3]{2}} \\ -\alpha, -\beta, -\gamma,$$ where $\varepsilon$ is a 3rd root of unity. So, L should be $\mathbb{Q} (\alpha, \beta, \gamma)$.
I can't understand if i can simplify the writing of L; i found that $\beta,\gamma\notin\mathbb{Q}(\alpha)$, because $\mathbb{Q}(\alpha)\subseteq\mathbb{R}$, but i can't go further than this. Actually, i had to study subfields of $\mathbb{Q}(\alpha)$, and the only tool i have is Galois theory, so i tried this way, but maybe there is a better one, let me know!