Question

Form the equation of lowest dimensions with rational coefficients, one of whose roots is
$-\sqrt{2}-\sqrt{-2}$.

Answer 1

$-\sqrt{2}-\sqrt{-2}=-\sqrt{2}-\sqrt{2}i$

The equation can have following roots

$-\sqrt{2}-\sqrt{2}i,-\sqrt{2}+\sqrt{2}i$ and $\sqrt{2}-\sqrt{2}i,\sqrt{2}+\sqrt{2}i$

Equation corresponding to first set is

$\{x-(-\sqrt{2}-\sqrt{2}i\left)\right\}\{x-(-\sqrt{2}+\sqrt{2}i\left)\right\}=0\left\{\right(x+\sqrt{2})+\sqrt{2}i\}\left\{\right(x+\sqrt{2})-\sqrt{2}i\}=0{(x+\sqrt{2})}^2-{\left(\sqrt{2}i\right)}^2=0x^2+2\sqrt{2}x+2+2=0x^2+2\sqrt{2}x+4=0$

Equation corresponding to second set is

$\{x-(\sqrt{2}-\sqrt{2}i\left)\right\}\{x-(\sqrt{2}+\sqrt{2}i\left)\right\}=0\left\{\right(x-\sqrt{2})+\sqrt{2}i\}\left\{\right(x-\sqrt{2})-\sqrt{2}i\}=0{(x-\sqrt{2})}^2-{\left(\sqrt{2}i\right)}^2=0x^2-2\sqrt{2}x+2+2=0x^2-2\sqrt{2}x+4=0$