Proposition 2

— The field of symmetric rational fractions in $X_1, \ldots, X_n$ is the field of fractions of the ring of symmetric polynomials in $X_1, \ldots, X_n$..

Let $f \in K(X_1, \ldots, X_n)$ be a symmetric rational fraction, and let $u_1, v_1$ be two elements of $K[X_1, \ldots, X_n]$ such that $f = \frac{u_1}{v_1}$. We put $v = \prod_{\sigma \in S_n} \psi_\sigma(v_1) \in K[X_1, \ldots, X_n]$ and $u = fv \in K[X_1, \ldots, X_n]$; then $v$ is symmetric, hence $u$ is symmetric, because $f$ is, and $f = \frac{u}{v}$, whence the result.