Ring of invariants of symmetric group acting on ring of formal power series

Question

Let $k$ be a field and $R=k[[x_1, \dots, x_n]]$ be the ring of formal power series in $n$ variables. Let $S_n$ be the symmetric group of order $n!$. Then $S_n$ acts on $R$ by $k$-automorphisms by permuting the coordinates. Let $R^{S_n}$ be the subring of $R$ of all elements that are fixed by the action of $S_n$. How can I prove that $R^{S_n}$ is regular, i.e., that it has Krull dimension $n$? The author claims that this follows from the fact that any symmetric polynomial in $A[x_1, \dots, x_n]$ over any ring $A$ can be expressed in a polynomial equation in $\sigma_1, \dots, \sigma_n$, where $\sigma_1, \dots, \sigma_n$ are the elementary symmetric polynomials. I don’t see how the statement follows from this fact.