An Irrational Non-Quadratic Root

Prove:

1. The equation $x^3-x-1=0$ has exactly one real root $\omega$, and $\omega$ is irrational. 2. $\omega$ is not a root of any quadratic equation $ax^2+bx+c=0$ with integer coefficients ($a,b,c\in\mathbb Z$, $a\neq 0$).