Symmetric Rational Functions vs Symmetric Polynomials

Question

Let $\{z,x_1,\ldots,x_n\}$ be variables and let $e_1,\ldots,e_n\in\mathbb{Q}[x_1,\ldots,x_n]$ be the elementary symmetric polynomials defined by $$z^n-e_1z^{n-1}+\cdots +(-1)^ne_n =(z-x_1)(z-x_2)\cdots (z-x_n).$$ Suppose that a polynomial $f(x_1,\ldots,x_n)\in\mathbb{Q}[x_1,\ldots,x_n]$ can be expressed in the form $$f(x_1,\ldots,x_n)=\frac{g(e_1,\ldots,e_n)}{h(e_1,\ldots,e_n)}$$ for some polyomials $g,h\in\mathbb{Q}[x_1,\ldots,x_n]$. How can I use this to prove that $$f(x_1,\ldots,x_n)=p(e_1,\ldots,e_n)$$ for some polynomial $p(x_1,\ldots,x_n)\in\mathbb{Q}[x_1,\ldots,x_n]$?