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Let $\mathbb{Z}[x_1,x_2,\ldots, x_n]$ be ring in $x_1,\ldots, x_n$ with coefficients from $\mathbb{Z}$.

Let $e_1=x_1+x_2+\cdots + x_n$, $e_2=\sum_{i<j} x_ix_j$, $\cdots$, $e_n=x_1x_2\cdots x_n$ : elementary symmetric polynomials.

Let $s_1=x_1+x_2+\cdots + x_n$, $s_2:=x_1^2+x_2^2+\cdots + x_n^2$, $\cdots$, $s_k=x_1^k+x_2^k+\cdots + x_n^k$.

If $R=\mathbb{Z}[e_1,e_2,\ldots, e_n]$ is subring of $\mathbb{Z}[x_1,x_2,\ldots, x_n]$ generated by $e_1,e_2,\ldots, e_n$ and $S=\mathbb{Z}[s_1,s_2,\ldots, ]$ is subring of $\mathbb{Z}[x_1,x_2,\ldots, x_n]$ generated by $s_1,s_2,\ldots, $, then is there any relation between $R$ and $S$?

(Note: I want to stay within $\mathbb{Z}[x_1,x_2,\ldots, x_n]$, i.e., I do not want to go for rational coefficients for symmetric polynomials).

Maths Rahul
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  • This is probably what you're looking for: https://en.wikipedia.org/wiki/Newton%27s_identities – Daniel Hast Feb 19 '22 at 15:04
  • @Daniel: This shows that $s_1,s_2,\ldots, s_n$ are polynomials in $e_1,e_2,\ldots, e_n$ with integer coefficients. However, $e_1,e_2,\ldots, e_n$ are polynomials in $s_1,s_2,\ldots$ but require rational coefficients.; right? – Maths Rahul Feb 19 '22 at 15:13
  • Exactly. More precisely, writing $e_1, \dots, e_n$ as polynomials in $s_1, \dots, s_n$ requires coefficients in $\mathbb{Z}[1/n!]$. – Daniel Hast Feb 19 '22 at 18:15

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