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?

For any integer $a$ except $0$ or $1$,  the polynomial $x^3 + a x^2 + a x + a$ has no rational roots.  Any rational root $A$ would have to be an integer (by Gauss's lemma, or the Rational Root Theorem). Now $A^3 + a A^2 + a A + a = 0 $ means 
$$a = - \frac{A^3}{A^2 + A + 1} = -A + 1 - \frac{1}{A^2 + A + 1}$$
which, if $A$ is an integer, is not an integer unless $A = 0$ (corresponding to $a=0$) or $A = -1$ (corresponding to $a = 1$): otherwise $A^2 + A + 1 = (A + 1/2)^2 + 3/4 > 1$.
