I have the following exercise:
Let $\mathbb{C}\left(e_{1}, e_{2}, e_{3}\right) \subset \mathbb{C}\left(x_{1}, x_{2}, x_{3}\right)$ be the subfield of symmetric functions where
$e_{1}=x_{1}+x_{2}+x_{3}, \quad e_{2}=x_{1} x_{2}+x_{1} x_{3}+x_{2} x_{3}, \quad e_{3}=x_{1} x_{2} x_{3}$
Find the degree of the extension of $\mathbb{C}\left(e_{1}, e_{2}, e_{3}\right)$ generated by the element $x_{1}+2 x_{2}+3 x_{3} \in \mathbb{C}\left(x_{1}, x_{2}, x_{3}\right)$.
I think that I should use the fact that the Galois group of $\mathbb{C}\left(x_{1}, x_{2}, x_{3}\right)/\mathbb{C}\left(e_{1}, e_{2}, e_{3}\right)$ is $S_{3}$ and probably the tower law for the intermediate extension, but that does not lead me anywhere. I have not seen something similar before, so any help would be much appreciated.
Let $E = ℂ(x_1, x_2, x_3)$, $F = ℂ(e_1, e_2, e_3)$ and $G = \operatorname{Gal} (E/F) \cong S_3$.
The intermediate fields of $E/F$ are the fixed fields $E^H$ corresponding to subgroups $H$ of $G$, so $ℂ(α) = E^H$ for some subgroup $H ⊆ G$.
What are the permutations of $G$ that fix $α = x_1 + 2x_2 + 3x_3$? Writing $ℂ(α) = E^H$ for some subgroup $H ⊆ G$, what can you conclude about $ℂ(α)$?