I am trying to find the splitting field $L$ of $x^3-8 \in \mathbb Q[x]$ and the degree of the extension $\mathbb Q \subset L$, computing Gal$(L/\mathbb Q)$ and finding the correspondence between the subgroups of it and the intermediate field extensions. Also I have to show that which intermediate extensions are normal in $\mathbb Q$.
I tried to solve it, the spliiting field is $\mathbb Q(i\sqrt3)$ and the degree of the extension is 2. How to proceed further? Specially about the correspondence?
1 and $i\sqrt3$ forms a basis of $L$ over $\mathbb Q$. so every element of $L$ has the form $a+bi\sqrt3$ If $f \in Gal(L/\mathbb Q)$ then $f(a+bi\sqrt3)=a+bf(i\sqrt 3)$ Now $(f(i\sqrt3))^2=f(-3)=-3 \implies f(i\sqrt3)=i\sqrt3$ Is it correct and the Galois group is ?