Let $f(x) \in K[x]$ be a monic polynomial, $L$ a splitting field for $f, \alpha_{1}, \ldots, \alpha_{n}$ the roots of $f$ in $L$. Let $\triangle(f)=\prod_{i<j}\left(\alpha_{i}-\alpha_{j}\right)^{2}$ be the discriminant of $f$. Explain why $\triangle(f)$ is a polynomial function in the coefficients of $f$, and determine $\triangle(f)$ when $f(x)=x^{3}+p x+q$.

Compute the Galois group of the polynomial $x^{3}-3 x+1 \in \mathbb{Q}[x]$.