I'm trying to factorize a polynomial over $\mathbb Q(i,\sqrt 5)$:
$f(x)= x^4 -4x^2 - 5$.
I've factorised it thus:
in the form $(x^2 + ax + b)(x^2 + cx + d)$ where $b =d = \sqrt{-5}$ and $a=-c=\sqrt{2\sqrt{-5}+4}$ and I've then factorized each of these quadratics further to the forms $(x+a)(x+b)$ so that the overall factorization of the original polynomial is $(x-i)(x+i)(x-\sqrt 5)(x+\sqrt 5)$.
When I multiply the above out to get quadratics again to check my answers, the $x$ coefficient I get is $(i+\sqrt 5)$. Could someone please tell me why $(i+\sqrt 5)$ is equal to $\sqrt 2{\sqrt{-5} + 4}$? Or have I done this factorisation in an incorrect way, though multiplying out everything seems to work out fine.