I'm trying to prove that $x=2^{1/3}$ cannot be the root of a non-zero second degree polynomial of $\mathbb{Q}[X]$.
I would see how to do it if I knew that either $a$, $b$, or $c$ is $0$, using a similar technique used to show that $\sqrt{2}$ is irrational for instance, but here I tried to write $ax^2+bx+c=0$, and to square / cube this expression by moving terms around, but I cant find a way to do it, I always find myself with $x$, $x^2$ and a rational number at the end.