Determine if there exist rational number a and irrational number A such that $A^3+aA^2+aA+a=0$. Determine if there exist a rational number a and irrational number A such that $A^3+aA^2+aA+a=0$. If so, can we say something about them? Are there infinitely many of them?

If you don't like $A=i, a=1$, you can work with $a=-n$, $n>1$ square-free integer.  By Eisenstein criterion, the polynomial $x^3-nx^2-nx-n$ is irreducible over $\mathbb{Q}$, so the positive root $A$ is irrational.
