Dummit and Foote, Section 14.6: "If the roots of a polynomial $f(x)$ are independent indeterminates over a field $F$, then so are the coefficients of $f(x)$." This is meant to complete the converse of the iff statement regarding this claim so that the two definitions of the general polynomial over $F$ in $n$ variables (i.e. having either indeterminate roots or indeterminate coefficients) are equivalent, but I'm stuck.
If the coefficients $s_1,s_2,...,s_n$ satisfy a polynomial relation over $F$ by $p(s_1,s_2,...,s_n)=0$ then since $s_i$ is the $i$th elementary symmetric function on the roots $x_1,x_2,...,x_n$ of $f(x)$ we can view it as a composite polynomial relation in $x_1,x_2,...,x_n$, but I don't know how to show this is nonzero and leads to a contradiction. Am I on the right track?