# Degree of a splitting field over $\mathbb{Q}$

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

Hint: $1-\sqrt{3}\in\Bbb Q(\sqrt{1+\sqrt{3}})$.
Note your field is $\mathbb{Q}$($\alpha$, $\beta$) where $\alpha$ and $\beta$ are your explicit roots and note then that $\alpha^2$ + $\beta^2$ = 2. Apply now the multiplication of degrees knowing that each of them is of degree 4 over $\mathbb{Q}$ and the other is of degree 2 over the first extension, by the above equation of sum of squares.