Let $f(x)=x^4-6x^2+4 \in \mathbb{Q}[x]$ and $L$ a splitting field of $f(x)$ over $\mathbb{Q}$.
How to determine the Galois group?
My idea was:
$x^4-6x^2+4=(x^2)^2-6x^2+4$
So $x^2 \in \lbrace \frac{6}{2} \pm \sqrt{(\frac{6}{2})^2-4} \rbrace = \lbrace 3+\sqrt{5},3-\sqrt{5} \rbrace$
Then $x \in \lbrace \sqrt{3+\sqrt{5}}, \sqrt{3-\sqrt{5}} \rbrace$
These are two roots of the polynomial. Now I'm not sure what to do next. I thought about finding the other two roots, but I don't know how.
How to continue? Or is there a better method to find the Galois group here?
