ac.commutative algebra - Do n-th Witt polynomials generate {P | P' is divisible by n} ?

EDIT: Proved it on my own. It easily follows from the Witt integrality theorem. Sorry for posting.

Let $Pinmathbb{Z}left[Xiright]$ be a polynomial (where $Xi$ is a family of symbols that we use as indeterminates, for instance $Xi=left(X_1,X_2,X_3,...right)$). Let $ninmathbb{N}$.

Prove or disprove that $displaystylefrac{delta}{deltaxi}Pin nmathbb{Z}left[Xiright]$ for every $xiinXi$ if and only if there exist polynomials $P_dinmathbb{Z}left[Xiright]$ for all divisors $d$ of $n$ such that $displaystyle P=sum_{dmid n}dP_d^{n/d}$.

A few remarks on this: The $Longleftarrow$ direction is trivial. I can prove the $Longrightarrow$ if $n$ is a prime power.

PS. No, this does not help in proving the Witt integrality theorem, even if it is true.