Determine if there exist rational number a and irrational number A such that $A^3+aA^2+aA+a=0$.

Question

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?

Answer 1

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 $A^2 + A + 1 = (A + 1/2)^2 + 3/4 > 1$.

Answer 2

The polynomial $X^3+pX^2+pX+p$ has no root if $p$ is a prime number, for example (the only potential roots are $1,-1,p,-p$. Check they are not).

But this polynomial has at least one real root $A$, since it has degree 3.

Answer 3

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.