Let \(R\) be a commutative ring and let \(x_1, \ldots, x_n\) be indeterminates over \(R\). The elementary symmetric functions in \(x_1, \ldots, x_n\) are \(e_0 \coloneqq 1\) and \[ e_k \coloneqq \sum_{1\leq i_1 < i_2 < \cdots < i_k\leq n} x_{i_1} x_{i_2}\cdots x_{i_k} \in R[x_1, \ldots, x_n], \qquad 1 \leq k \leq n. \]
In \(R[x_1,\ldots,x_n][t]\), \[ \prod_{i=1}^n (t - x_i) = \sum_{k=0}^n (-1)^k e_k t^{n-k}. \] Consequently, if a monic polynomial of degree \(n\) over a commutative ring factors as \(\prod_{i=1}^n(t-r_i)\), its coefficient of \(t^{n-k}\) is \((-1)^k e_k(r_1,\ldots,r_n)\).
Let \(R\) be a commutative ring. Every symmetric polynomial in \(R[x_1, \ldots, x_n]\) is equal to \(g(e_1, \ldots, e_n)\) for a unique polynomial \(g\in R[y_1,\ldots,y_n]\).
See [@DF04].