Hey I want to check if I have done right the following exercises:
a) Show that the polynomial $P(X)=X^3-2$ is irreducible over $\mathbb{Q}$.
Here I used Eisenstein Criterion with $p=2$.
b) Give the splitting field $L$ of $P(X)$.
Here I wrote $L=\mathbb{Q}(\sqrt[3]{2}, e^{2\pi i/3})$
c) Determine $[L:\mathbb{Q}]$.
$[L:\mathbb{Q}]=[L:\mathbb{Q}(\sqrt[3]{2})][{Q}(\sqrt[3]{2}):\mathbb{Q}] =6$
d) Explain why $L/\mathbb{Q}$ is Galois. Determine the Galois group of the polynomial P(X) and the isomorphism type.
Since $L$ is a splitting field of $P(X)$ and $\text{char}(\mathbb{Q})=0 $ it is Galois.
Since $[L:\mathbb{Q}]=|\text{Gal}(L/\mathbb{Q})|$, we have to find 6 automorphisms that permute the roots of $P(X)$.
Here is a table: 
So we have $Gal(L/\mathbb{Q}) \cong S_3$.
e)Specify all intermediate fields of the extension $L/\mathbb{Q}$.
Using d) here are the intermediate fields: $\mathbb{Q}(\sqrt[3]{2}),\mathbb{Q}(\sqrt[3]{2} e^{2\pi i/3}),\mathbb{Q}(\sqrt[3]{2} (e^{2\pi i/3})^2), \mathbb{Q}( e^{2\pi i/3})$
Have I done everything correctly or is there an error somewhere?
