Exercise 4, § 5

For all $n \in \mathbf{N}$, let $\binom{T}{n}$ denote the polynomial

† For more details on this exercise, see K. W. Gruenberg, Lecture Notes in Math., no. 143, Chapter 3, and R. Swan, J. of Algebra, 12 (1969), pp. 585–601.

If $f(T)$ is a polynomial, let $\Delta f$ denote the polynomial $f(T + 1) - f(T)$. Then $\Delta \binom{T}{0} = \Delta 1 = 0$ and $\Delta \binom{T}{n} = \binom{T}{n-1}$ if $n \geq 1$.

(a) Let $f \in \mathbf{Q}[T]$. Show the equivalence of the following properties: (i) $f$ maps $\mathbf{Z}$ into itself. (ii) $f$ is a linear combination with coefficients in $\mathbf{Z}$ of the $\binom{T}{n}$. (iii) $f(0) \in \mathbf{Z}$ and $\Delta f$ maps $\mathbf{Z}$ into itself. (Argue by induction on $\deg(f)$, noting that $\deg(\Delta f) = \deg(f) - 1$ if $\deg(f) \neq 0$.) A polynomial $f$ satisfying the above properties is called binomial. The sum, product and composition of two binomial polynomials is a binomial polynomial.

(b) Let $p$ be a prime number and let $f$ be a binomial polynomial. Show that $f$ maps the ring $\mathbf{Z}_p$ of $p$-adic integers into itself (use the continuity of $f$ and the fact that $\mathbf{Z}$ is dense in $\mathbf{Z}_p$).

(c) Let $P$ be a set of prime numbers, let $S_P$ be the set of $P$-integers (§ 4, Exercise 14) and let $\mathbf{Z}_P = S_P^{-1}\mathbf{Z}$. Show that if $f$ is a binomial polynomial, $f$ maps $\mathbf{Z}_P$ into itself (apply (b) to the prime numbers $p$ not belonging to $P$).