I'm reading a book on trigonometric series, and the author uses the following result in a proof:
Let $f\in \mathbb{Z}[x]$ be a polynomial (with integers coefficients) of degree $n>0$, and let $S\in\mathbb{Q}[x_1,\dots,x_n]$ be a symmetric polynomial.
Let $\alpha_1,\dots,\alpha_n$ be the roots of $f$. Then $S(\alpha_1,\dots,\alpha_n)\in \mathbb{Q}$.
Is there an elementary proof of this fact? I've tried writing
$$\alpha_k=\sum_{i\neq k}a_i\alpha_k^i, \ 1\le k\le n$$
So
$$S(\alpha_1,\dots,\alpha_n)=S\left(\sum_{i\neq 1}a_i\alpha_1^i,\dots,\sum_{i\neq n}a_i\alpha_n^i\right)$$
which does not seem too useful. Also, by the fundamental theorem of symmetric polynomials, it would suffice to show the claim for elementary symmetric polynomials.