How to find the splitting field of $X^4-10X^2+1$ ? I found the roots
\begin{align*} X^4-10X^2+1=0&\iff (X^2-5)^2-24=0\\ &\iff X^2-5=\pm 2\sqrt 6\\ &\iff X^2=5\pm 2\sqrt 6\\ &\iff X\in\left\{\sqrt{5+2\sqrt 6},\sqrt{5-2\sqrt 6},-\sqrt{5+2\sqrt 6},-\sqrt{5-2\sqrt 6}\right\} \end{align*}
But I'm not able to continue...
Hint: $(\sqrt 3 + \sqrt 2)^2 = 5 + 2\sqrt6$ and $(\sqrt 3 - \sqrt 2)^2 = 5 - 2\sqrt 6$, so your splitting field contains $\sqrt 2$ and $\sqrt 3$.