Let $F=\mathbb{C}\left(X_{1}, \ldots, X_{n}\right)$ be a field of rational functions in $n$ variables over $\mathbb{C}$, and let $s_{1}, \ldots, s_{n}$ be the elementary symmetric polynomials:

$$s_{j}:=\sum_{\left\{i_{1}, \ldots, i_{j}\right\} \subset\{1, \ldots, n\}} X_{i_{1}} \cdots X_{i_{j}} \in F \quad(1 \leqslant j \leqslant n),$$

and let $K=\mathbb{C}\left(s_{1}, \ldots, s_{n}\right)$ be the subfield of $F$ generated by $s_{1}, \ldots, s_{n}$. Let $1 \leqslant m \leqslant n$, and $Y:=X_{1}+\cdots+X_{m} \in F$. Let $K(Y)$ be the subfield of $F$ generated by $Y$ over $K$. Find the degree $[K(Y): K]$.

[Standard facts about the fields $F, K$ and Galois extensions can be quoted without proof, as long as they are clearly stated.]