Exercise 3, § 1

Let $A$ be a commutative ring. For every polynomial $f \in A[X]$ we write in the polynomial ring $A[X, Y]$ of polynomials in two indeterminates over $A$,

where $\Delta$ is thus an $A$-linear mapping of $A[X]$ into itself. a) If $Z$ is an indeterminate and $\tau_Z$ the $A$-linear mapping of $A[X]$ into $A[X, Z]$ such that $\tau_Z f = f(X + Z)$, show that $\tau_Z \circ \Delta = \Delta \circ \tau_Z$ (where on the right $\Delta$ is the mapping defined above in the ring $B[X]$, where $B = A[Z]$). Deduce that in $A[X]$ we have

for integers $m, n \geq 0$ (cf. IV, p. 8). b) Suppose that $A$ is a ring of characteristic $p > 0$ and for each integer $k \geq 0$ put $D_k = \Delta_{p^k}$. Show that if $f \in A[X]$ and $g \in A[X^{p^k}]$ then $D_k(fg) = g \cdot D_k f + f \cdot D_k g$. Every polynomial $f \in A[X]$ can be uniquely written in the form $f(X) = \sum_{m=0}^{\infty} X^{mp^k} g_m(X)$ where $\deg(g_m) < p^k$; show that

and deduce that $D_k^p = 0$.

c) For every integer $m > 0$ let $m = \alpha_0 + \alpha_1 p + \ldots + \alpha_k p^k$ be the expansion of $m$ in base $p$ (Set Theory, III, p. 177), where thus $0 \leq a, < p$ for $0 \leq j \leq k$; suppose that $a_j \neq 0$. Show that

(If $\Delta'_m$ is the right-hand side of this relation, note that if we introduce the expansion $f(X + Y) = \sum_{q=0}^{\infty} f_q(Y) X^q$, we have $\Delta'_m f(X + Y) = \sum_{q=0}^{\infty} f_q(Y) \Delta'_m(X^q)$ and deduce that this may be written $\Delta'_m f(X + Y) = f_m(X) + Yg(X, Y)$, where $g$ is a polynomial.)