Let $f \in k(x_1, ..., x_n)$, where $k$ is a field
$Gal(k(x_1, ..., x_n) / k_{sym}(x_1, ..., x_n)) \cong S_4$
$|\{\phi_\sigma(f)\ |\ \sigma \in S_4 \}| = 2$ where $\phi$ is orbit of the function $f$. Let $q_1, q_2 \in k(\sigma_1, ..., \sigma_n)$
if char(k) $\neq$ 2
Then $f = q_1\ +\ q_2V_n$, where $V_n$ - Vandermonde Matrix
And if $char(k) = 2$
Then $f = q_1\ +\ q_2F$, where $F = \Sigma_{\sigma \in A_4} \Pi_{\sigma(i)}x^{i-1}$
So, i seen this solution: Prove that $f = q_1 + Gq_2$ for some $q_1, q_2 \in \mathbb{k}_{sym}(x_1,\dots,x_n)$ but I don't understand how I can prove second and third point. I think that I need to use theorems about Galois Resolvent, but I don't known how.


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