Please help me to answer the following problem:
Let $f(x)=x^3-6x-2\in\mathbb{Q}[x]$ and $L$ be a splitting field of $f(x)$ over $\mathbb{Q}$.
$f$ is irreducible in $\mathbb{Q}[x]$ and its discriminant is $756$. Up to isomorphism $\operatorname{Gal}(L/\mathbb{Q})=S_3$
Find an explicit expression of a root of $f(x)$ in term of radicals and use it to write a repeated radical extention $\mathbb{Q}⊂K$ such that $f(x)$ splits in $K$.