Galois field extension and number of intermediate fields Given the galois extension $L/K$, where $L=\mathbb{Q}(\zeta, \sqrt[3]{2})$. I've already calculated that $[L:\mathbb{Q}]=6$. I'm trying to prove that the number of intermediate fields is also 6, including $L$ and $\mathbb{Q}$ as intermediate fields. I'm told that the fundamental theorem of galois theory can be useful here, but I can't find a use for it.
 A: Hint:
What is the Galois group $\operatorname{Gal}(L/\mathbb{Q})$?  
Well, the minimal polynomial for which $L$ is a splitting field is $f(x) = x^3 - 2$, which has roots $\sqrt[3]{2}, \zeta \sqrt[3]{2}$, and $\zeta^2 \sqrt[3]{2}$.  It is useful to remember that each element of the Galois group is uniquely determined by its action on the roots of $f$.
Now, complex conjugation is always a legitimate $\mathbb{Q}$-automorphism.  That is, $\phi(\zeta) = \overline{\zeta}$.  Further, I claim that another automorphism $\tau$ is defined by $\sqrt[3]{2} \mapsto \zeta\sqrt[3]{2}$.  Can you prove this?
What happens when we take powers of $\phi$ and $\tau$ and compose them with each other?  For me, it is especially illuminating to view the roots of $f$ as vertices of a triangle in the complex plane and see how these automorphisms are acting on it.

Alternative path:
It is a theorem that the Galois group will be a subgroup of the symmetric group acting on the roots of $f$.  Therefore, we know $\operatorname{Gal}(L/\mathbb{Q}) \leq S_3$.  However, we also know that $|\operatorname{Gal}(L/\mathbb{Q})| = [L:\mathbb{Q}] = 6$, so that certainly narrows down the possibilities for $\operatorname{Gal}(L/\mathbb{Q})$!

Once you've determined the Galois group, then what are its subgroups?  Remember the fundamental theorem tells us that there is a one-to-one correspondence between subgroups of the Galois group and intermediate fields of the Galois extension.
