My question is to find the splitting field of $x^4-2x^2-2$ over $\mathbb{Q}$.
By finding out the zeros, I find the root as $\sqrt{1+\sqrt{3}},-\sqrt{1+\sqrt{3}},\sqrt{1-\sqrt{3}},-\sqrt{1-\sqrt{3}}$.
I also find out that $\sqrt{2}i=\sqrt{1+\sqrt{3}}\sqrt{1-\sqrt{3}}$.
So, I concluded that splitting field would be $\mathbb{Q}(\sqrt{1+\sqrt{3}},\sqrt{2}i)$. However, I don't know the way to prove that it is the exact splitting field. Is there any way to prove it without guessing? Thank you so much!