Prove that $K$ is finite Galois over $\mathbb{Q}$ I just need a bit of quick help in understanding some solutions to a problem set. 
The question is this:
(a) Let $K=\mathbb{Q}(\alpha)$ with $\alpha$ a zero of $f(x) = x^3-3x+1$. Prove that $K$ is Galois over $\mathbb{Q}$.
I easily showed, $f(x)$ is irreducible, and detemined that is $\alpha$ s a root of $f(x)$, then so is $\alpha^2-2$. I was stuck, and the solutions says:
Let $\beta$ be a third zero of $f(x)$, then $\alpha + ( \alpha^2-2)+ \beta = 0$, since the coefficient of $x^2$ is zero in $f(x)$.
Why is this so? I am struggling to see this.

Have you heard of the discriminant? This is one way to show that K is Galois over the field of rational numbers.
