Question

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

Answer 1

Here we must have $\sqrt{2}+\sqrt{-3},\sqrt{2}-\sqrt{-3}$ as one pair of roots, and $-\sqrt{2}+\sqrt{-3},-\sqrt{2}-\sqrt{-3}$ as another pair.
Corresponding to the first pair we have the quadratic factor $x^2-2\sqrt{2}x+5$, and corresponding to the second pair we have the quadratic factor
$x^2+2\sqrt{2}x+5$.
Thus the required equation is
$(x^2+2\sqrt{2}x+5)(x^2-2\sqrt{2}x+5)=0$,
or $(x^2+5{)}^2-8x^2=0$,
or $x^4+2x^2+25=0$.