Frobenius substitution on cyclotomic power series ($\phi (f)(T)=f((1+T{)}^p-1)$)

On $R={\mathbb{Z}}_p[[T]]$, this injective substitution reduces modulo $p$ to $f(T)\mapsto f(T^p)$. The ring is a free module of rank $p$ over $\phi (R)$, with basis $1,(1+T),\dots ,(1+T{)}^{p-1}$. Prove this first modulo $p$, where the change of basis from $1,T,\dots ,T^{p-1}$ is triangular and invertible, and then lift coefficients successively modulo powers of $p$.

For $R={\mathbb{Z}}_p[[T]]$ and $\varphi (f)=f((1+T{)}^p-1)$, reduction modulo $p$ is the injective substitution $\overline{f}(T)\mapsto \overline{f}(T^p)$. Thus $\varphi (f)\in pR$ implies $f\in pR$. Dividing by $p$ and repeating proves the displayed equality for every $k\ge 1$. In particular, a ring congruence of Frobenius substitutions on cyclotomic power series reflects back to the original formal power series.

In the unique decomposition $f={\sum }_{i=0}^{p-1}(1+T{)}^i\phi (f_i)$, define $\psi (f)=f_0$. Averaging translates over the p-th roots of unity annihilates every nonzero basis index and retains the zeroth, proving the displayed identity and its integrality. Thus $\psi \phi =1$ and $\psi (\phi (g)f)=g\psi (f)$. Under the Amice transform, $\psi =0$ characterizes measures supported on ${\mathbb{Z}}_p^{\times }$.