abstract algebra - Finding the degree of the splitting field

Find the degree of the splitting field of $f(x)=x^3-3x-1$ over $\mathbb{Q}$. I know that this polynomial is irreducible by using Eisenstein's criteria(by letting first $x=y+1$), and for every cubic polynomial $f∈\mathbb{Q}[x]$, the splitting field of $f$ over $\mathbb{Q}$ is a radical extension. But how can I start my work? Thanks!

Answer

The degree is 3. Since $f$ is irreducible, $L=Q[x]/(f)$ is a separable field of degree $3$ since the characteristic of $Q$ is zero. $Gal(L:Q)$ has order 3 and is generated by $h$. Let $u$ be a root of $f$ in $L$ we can take $u$ to be the class of $x$ in $Q[x]/(f)$, $h(u)\neq u$ since $u$ is not in $Q$. $h(u),h^2(u)$ are also roots of $f$ and remark that $u,h(u), h^2(u)$ are distinct (example, $h(u)=h^2(u)$ implies $h(h(u))=h^3(u)=u, h^2(u)=u, h(h^2(u))=h(u)=u$ contradiction). This implies that $L$ is a splitting field and $f=(X-u)(X-h(u))(X-h^2(u))$.