This isn't as much a "How do I solve this assignment" question as much as it's a "how am I supposed to understand this concept", but I'm including the setup + first part of the assignment to give some context.
Let $K$ be a field and $f\in K[x]$ a separable polynomial of degree n. Let L be the splitting field of $f$, and $G = \text{Gal}(L/K)$. We denote the roots of $f$ in $L$ as $u_1,...,u_n$. For an element $g\in G$, we denote the corresponding permutation of the roots as an element in $S_n$ as $\pi_g$, so that $gu_i= u_{\pi_g (i)}$. Now, we define the polynomial
\begin{equation} F(x) = \prod_{\sigma\in S_n}(x-(u_{\sigma(1)}y_1+...+u_{\sigma(n)}y_n)) \end{equation}
We see that $F\in L[y_1, ..., y_n, x]$. The first subproblem, which I believe I've solved, is showing that $F\in K[y_1, ..., y_n, x]$. As near as I can tell, the coefficients of $F$ are elementary symmetric polynomials in the roots, and the coefficients of THOSE coefficients are elementary symmetric polynomials in $u_1,...,u_n$, solving the problem. However, the next subproblem is what confused me greatly.
Assume $F(x)$ has irreducible factorization into monic polynomials $F_1(x)...F_t(x)$, all factors in $K[y_1, ..., y_n, x]$. Assume $x-\sum_i u_{\sigma(i)}y_i$ is a factor of $F_1$ for some fixed permutation $\sigma$. Show that $F_1(x) = \prod_{g\in(G)}(x-\sum_i u_{\pi_g \sigma(i)}y_i)$.
This subproblem baffles me. Intuitively, this reminds me of the situation where if an irreducible polynomial has a root in a Galois extension, it splits completely in that extension. However, this is not a polynomial ring over a field, but a polynomial ring over a polynomial ring over... a field, for however many variables we have, so I'm not sure why that theorem would even apply. Furthermore, I have no idea what the connection between the Galois group $G$ and these polynomials is supposed to be. This is for an introductory Galois theory course, so I doubt there's actually a huge amount of theory about this I just haven't been taught yet. I just don't know how to apply my existing tools to this situation.