Question

Form the equation of lowest dimensions with rational coefficients, one of whose roots is
$\sqrt{3}+\sqrt{-2}$.

Answer 1

$\sqrt{3}+\sqrt{-2}=\sqrt{3}+\sqrt{2}i$

The equation mus be a quadratic as imaginary roots always exists in pairs

If $\sqrt{3}+\sqrt{2}i$ is one root then $\sqrt{3}-\sqrt{2}i$ is the other root

So the quadratic equation is

$\{x-(\sqrt{3}+\sqrt{2}i)\}\{x-(\sqrt{3}-\sqrt{2}i)\}=0\{(x-\sqrt{3})-\sqrt{2}i\}\{(x-\sqrt{3})+\sqrt{2}i\}=0{(x-\sqrt{3})}^2-{\left(\sqrt{2}i\right)}^2=0x^2-2\sqrt{3}x+3+2=0x^2-2\sqrt{3}x+5=0$