Question

Form the equation of lowest dimensions with rational coefficients, one of whose roots is
$-\sqrt{-1}+\sqrt{5}$.

Answer 1

$\sqrt{-1}+\sqrt{5}=-i+\sqrt{5}=\sqrt{5}-i$

The equation can have following roots

$\sqrt{5}-i,\sqrt{5}+i$ and $-\sqrt{5}-i,-\sqrt{5}+i$

Equation corresponding to first set is

$\{x-(\sqrt{5}-i\left)\right\}\{x-(\sqrt{5}+i,\left)\right\}=0\left\{\right(x-\sqrt{5})+i\}\left\{\right(x-\sqrt{5})-i\}=0{(x-\sqrt{5})}^2-i^2=0x^2-2\sqrt{5}x+6=0$

Equation corresponding to second set is

$\{x-(-\sqrt{5}-i\left)\right\}\{x-(-\sqrt{5}+i,\left)\right\}=0\left\{\right(x+\sqrt{5})+i\}\left\{\right(x+\sqrt{5})-i\}=0{(x+\sqrt{5})}^2-i^2=0x^2+2\sqrt{5}x+6=0$