Power sum symmetric polynomials

Definition

Let $R$ be a commutative ring and let $n \ge 1$. For $k \ge 1$, the $k$-th power sum in $x_1, \dots, x_n$ is $$p_k = x_1^k + x_2^k + \dots + x_n^k \in R[x_1, \dots, x_n].$$

Remarks

Each $p_k$ is symmetric and homogeneous of degree $k$. Newton's identities relate the power sums to the elementary symmetric polynomials. If $R$ contains $\mathbb{Q}$, then every symmetric polynomial in $R[x_1, \dots, x_n]$ is a polynomial in $p_1, \dots, p_n$. This fails over $\mathbb{Z}$: $e_2 = \tfrac{1}{2}(p_1^2 - p_2)$ needs the coefficient $\tfrac{1}{2}$.

Explanations

L1 human

The power sums add up the same power of every variable. They are symmetric, and over the rationals they can replace the elementary symmetric polynomials as generators.