I found that the roots of $f(x) = x^4+10x^2+5$ are $\pm i\sqrt{5-2\sqrt{5}}$ and $\pm i\sqrt{5+2\sqrt{5}}$, then the splitting field of this polynomial over the rationals is $K := \mathbb{Q}{(i\sqrt{5-2\sqrt{5}},i\sqrt{5+2\sqrt{5}})}$.
However, I have to prove that the degree of $K/\mathbb{Q}$ is 4, and that happens only if $K=\mathbb{Q}(i\sqrt{5-2\sqrt{5}})$, so in this case I need to find out that $i\sqrt{5+2\sqrt{5}}\in \mathbb{Q}(i\sqrt{5-2\sqrt{5}})$, but I cannot show this by field operations.
Can anyone give me a hint to do this? Thanks!