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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.

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    $x^4-4x^2-5=(x^2-2)^2-9=((x^2-2)+3)((x^2-2)-3)=(x^2+1)(x^2-5)=(x+i)(x-i)(x+\sqrt{5})(x-\sqrt{5})$.2012-04-22
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    Also, $(\sqrt{5}+i)^2=5+2i\sqrt{5}-1=4+2\sqrt{-5}$.2012-04-22
  • 0
    Of course..! Thank you for the point out :)2012-04-22

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