I'm trying to solve the following problem: Let
\begin{equation*} f(x) = x^4 - 2x^2 - 2 \in \mathbb{Q}[x] \end{equation*}
and $ E $ be its splitting field. What is the degree $ [E: \mathbb{Q}] $?
First of all, the roots of this polynomial are $ \pm \sqrt{1 + \sqrt{3}}, \pm \sqrt{1 - \sqrt{3}} $. The first two are real and the other two are complex. It looks as if $ E $ was equal to $ \mathbb{Q}(\sqrt{1 + \sqrt{3}}, \sqrt{1 - \sqrt{3}}) $. The degree of these elements over $ \mathbb{Q} $ is $ 4 $ since $ f(x) $ is irreducible by Eistenstein's criterion. However, I can't tell what the degree of $ \sqrt{1 - \sqrt{3}} $ over $ \mathbb{Q}(\sqrt{1 + \sqrt{3}}) $ is, since I don't know how to check whether $ f $ is irreducible over this field.
I'd appreciate any ideas on how to get to that