I'm currently reading the paper "On the $\mu$-invariant of the Γ-transform of a rational function" by W Sinnott. In this paper, he gives an alternate proof that $\mu=0$ for abelian number fields. (This was originally proven by Ferrero–Washington.)
The heart of the proof is an ingenious result in commutative algebra. This result, which I've copied below, allows Sinnott to bypass the more analytic arguments of normal numbers used in the original proof of the Ferrero-Washington Theorem, using a instead a purely algebraic approach.
Let $\mu_{p-1}$ denote the $(p-1)$-st roots of unity in the ring $\mathbf{Z}_p$ of $p$-adic integers. Suppose that for each root of unity $\alpha \in \mu_{p-1} \subseteq \mathbf{Z}_{p}$, we are given a rational function $F_{\alpha}(T) \in \mathbf{Z}_p[[T]]$. If $$\sum_{\alpha\in \mu_{p-1}} F_{\alpha}((1+T)^{\alpha}-1) \equiv 0 \mod p\mathbf{Z}_{p}[[T]], $$ then for all $\alpha \in \mu_{p-1}$, we have $$F_{\alpha}(T)-F_{-\alpha}((1+T)^{-1}-1) \equiv \text{constant} \mod p\mathbf{Z}_{p}[[T]].$$ The constant on the right hand side is allowed to depend on $\alpha$.
This is purely a theorem in commutative algebra. My question is: is the above lemma known to hold true if we remove the assumption that $F_{\alpha}(T)$ are rational functions? That is, does the lemma still hold if the $F_{\alpha}(T)$ are arbitrary elements of $\mathbf{Z}_p[[T]]$?
