Let $\mathbb C [x_1, \dots, x_n]$ be a ring of polynomials with complex coefficients. We then define the symmetric polynomials \begin{align*} S_0 &= 1 \\ S_1 &= x_1 + \cdots + x_n \\ S_2 &= x_1x_2 + x_1 x_3 + \cdots + x_{n-2}x_n+ x_{n-1}x_n \\ \cdots \\ S_n &= x_1x_2\cdots x_n \end{align*} and the symmetric $k$-power polynomials, for every $k\in \mathbb N$, \begin{align*} P_{k, 0} &= 1 \\ P_{k, 1} &= x_1^k + \cdots + x_n^k \\ P_{k, 2} &= x_1^kx_2^k + x_1^k x_3^k + \cdots + x_{n-2}^k x_n^k + x_{n-1}^k x_n^k \\ \cdots \\ P_{k, n} &= x_1^k x_2^k \cdots x_n^k \end{align*}
Newton's identities give us \begin{equation} P_{k,1} = \sum_{i=1}^k (-1)^{i-1} S_i P_{k-i,1} \end{equation} which implies that $P_{k,1}$ can be written as a polynomial of $S_1, \dots, S_n$ with integral coefficients. My question is
Is it possible to write $P_{k,i}$ as a polynomial of $S_1, \dots, S_n$ with integral coefficients?