<s>[INST] How to find the splitting field of $X^4-10X^2+1$? 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... 
 [/INST] 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 $\sqrt 3$.
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