Formal power series on discrete valuation rings

Question

Let's start with a simple result: For $F(x)\in \mathbb{Z}_p[x]$,
$$F(x)^p\equiv F(x^p) \mod p $$ I generalize this result to a problem: Let $K$ be a local field, $A$ be its ring of integers with maximal ideal $\mathfrak{m}$ generated by $\pi$, $k$ be its residue field and $q:=\# k$. Let $F(x_1,\dots,x_n)\in A[[x_1,\dots,x_n]]$ be a formal power series over $A$. Do we have a similar result: $$F(x_1,\dots,x_n)^q\equiv F(x_1^q,\dots,x_n^q) \mod \pi$$

Answer 1

Sure, let $q=p^m$,

$R=A/(\pi)[[x]]$ is a ring of characteristic $p$, whence for any $F(x)\in R$, $$F(x)^{p^m}= F^{p^m}(x^{p^m})$$

where $F^{p^m}(..)$ means applying the $m$-th power Frobenius to the coefficients, and since the latter is the identity on $A/(\pi)$ we get $$F(x)^{p^m}=F(x^{p^m})$$