I've been doing a little bit of field theory for number fields but not much with function fields. The question originally asked says "For some field F, show that the field $F(u_1,\ldots, u_n)$ is a Galois extension of $F(s_1,\ldots,s_n)$ where the $s_i$ are the $i^{th}$ elementary symmetric polynomials in $n$ variables and show the Galois group is $S_n$."
Immediately I replaced $F(s_1,\ldots,s_n)$ with rational symmetric functions because of the fundamental theorem of symmetric polynomials. Then, it seems definitional almost that the Galois group is $S_n$ because I think the way symmetric polynomials are defined is by their invariance under the action of $S_n$ on the order of the variables. However I am confused about how to show this rigorously.
For now I feel like my biggest roadblock is understanding how to work with function fields. For number fields, I would make a formal polynomial like $X^2-2$ over the rationals and say this polynomial is irreducible and then adjoin $\sqrt{2}$ and this would be a Galois extension. It seems like if I were working with function fields of a single variable, $F(t)$ Galois extensions may look the same, $X^2-t$ and this would irreducible over the rational functions and I can adjoin $\sqrt{t}$ and this extension is Galois. But for function fields over several variables, does my formal polynomial have like several formal variables? Please help!