In Stewart and Tall's book on Algebraic Number Theory, they give a theorem of Newton:
Theorem 1.12. Let $R$ be a ring. Then every symmetric polynomial in $R[t_1, \ldots, t_n]$ is expressible as a polynomial with coefficients in $R$ in the elementary symmetric polynomials $s_1, \ldots, s_n$.
After proving this they give the following corollary.
Corollary 1.14. Suppose that $L$ is an extension of the field $K$, $p \in K[t]$, $\partial p = n$, and the zeros of $p$ are $\theta_1, \ldots, \theta_n \in L$. If $h(t_1,\ldots,t_n) \in K[t_1,\ldots,t_n]$ is symmetric, then $h(\theta_1, ..., \theta_n) \in K$.
Note: $\partial p$ denotes the degree of polynomial $p$.
My Question: I don't see how the corollary follows from the theorem in any straightforward way. Surely the way to apply the theorem is to first consider any symmetric polynomial $h(t_1,\ldots,t_n) \in K[t_1,\ldots,t_n]$ and then rewrite $h$ in terms of the elementary symmetric polynomials according to theorem 1.12. But I don't see why evaluating these elementary symmetric polynomials at roots $\theta_1,\ldots, \theta_n$ should always yield something in $K$.