TITLE: Show that a specific $w$ cannot be the root of an quadratic with integer coefficients.
QUESTION [6 upvotes]: Let $w$ be the only real root of $x^3-x-1=0$. Show that
 $w$  cannot satisfy the quadratic $ax^2 + bx + c$ ,where $a,b,c\in \Bbb Z$.
I have written 
$$w^3=w+1$$
but I can't go any further than this. Thank you.

REPLY [2 votes]: $p(x)=x^3-x-1$ is an irreducible polynomial over $\mathbb{Q}$, since it is irreducible over $\mathbb{F}_2$ (since it is a third-degree polynomial, proving that it is irreducible over a finite field is equivalent to proving that it has no roots in the field. But $p(x)$ is a trinomial, so $p(0)=p(1)=1$ in $\mathbb{F}_2$) . This gives that any root of $p(x)$ is an algebraic number of degree $3$ over $\mathbb{Q}$, so it cannot be the root of any quadratic polynomial with integer coefficients.