Exercise 47, § 1

Let $A$ be a ring. Define mappings $F_n, V_n (n \in \mathbf{J})$ of $\Lambda(A)$ into itself by the formulas

for every $a \in U(A)$.

a) One has $L(F_n(f)) = f_n(L(f))$ and $L(V_n(f)) = v_n(L(f))$ for every $f \in \Lambda(A)$. Let $n, m$ be in $\mathbf{J}$ and put $d = \operatorname{pgcd}(n, m)$.

b) Now prove that $F_n$ is an endomorphism of the ring $\Lambda(A)$, and that one has $F_n \circ F_m = F_{nm}$. For every prime number $p$ and every $f \in \Lambda(A)$, one has $F_p(f) \equiv f^{*p} \mod. p * \Lambda(A)$, where $f^{*p}$ denotes the product in the ring $\Lambda(A)$ of $p$ terms equal to $f$, and where $p * \Lambda(A)$ denotes the principal ideal of $\Lambda(A)$ generated by the sum in $\Lambda(A)$ of $p$ terms equal to the identity element $1 + T$, in other words by $(1 + T)^p$.

c) Prove that $V_n$ is an endomorphism of the additive group of $\Lambda(A)$, and that one has $V_n \circ V_m = V_{nm}$.

d) For every $f \in \Lambda(A)$, one has $F_n(V_m(f)) = V_{m/d}(F_{n/d}(f))^d$. In particular $F_n(V_n(f)) = f^n$, and $F_n \circ V_m = V_m \circ F_n$ if $d = 1$.

e) For every $f, g$ in $\Lambda(A)$, one has

In particular, one has and

f) One has

for every $f \in \Lambda(A)$. In particular for every $a \in A$.

g) For every $a \in A$, one has

For every $f \in \Lambda(A)$, one has where $N$ denotes the norm in the extension $A[[T^n]] \subset A[[T]]$. (It is enough to treat the case where $f \in A[T]$, then to embed $A$ in a ring $B$ where $f$ decomposes into a product of linear factors, to which one may apply the first formula.)

For every $a \in A$, one has

(Observe that $1 - a(-T)^m = V_m(1 + aT)$ and use $c$ and $f$.)

h) For every $a, b \in A$, one has

(Use the first formula for $e$.)